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MANNHEIM 


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MULTIPLEX 
SLIDE  RULES 


By 
t.  W.  ROSENTHAL 


J  PUBLISHED  BT 


EUGENE  DIETZGEN  CO. 

MANUFACTURERS  AND  IMPORTERS  OF 

DRAWING    MATERIALS 

181  Monroe  Street,  *  *  *  *  CHICAGO,  ILL. 
119-121  West  23d  Street,  *  NEW  YORK,  N.  Y. 
14  First:  Street,  *  *  *  SAN  FRANCISCO,  CAL. 
145  Bar  onne  Street,  *    *    NEW  ORLEANS,  LA. 


C.  HELLER 

CONSULTING   ENGiNESRl 
San    FhANClsuo,  Oau, 


O:  HELLER 

CONSULTING   ENGINE 

i>  a  -v    (-"rtANCl   ' 


MANNHEIM 

MULTIPLEX 
SLIDE    RULES 


Theory  and   Practical  Application 


By 
L.    W.    ROSENTHAL 

Elec.  Eng.  Assoc*  A.  I.  E.  E. 

Inventor  and  Patentee  of  the  Multiplex  Slide  Rule 


COPYRIGHT*   1905,    BY  EUGENE   DlErr,ZGUlS   Co. 


CONTENTS-PART    I. 

Mannheim  Slide  Rule. 


PAGE 

I.  INTRODUCTION. 

1..  Application 7 

2.  Qualifications 7 

3.  Accuracy 7 

4.  Saving  in  Time  and  Labor  8 

II.  THEORY   OF  LOGARITHMS. 

5.  Definition 8- 

6.  Common  Logarithms ....  8 

7.  Relation    between    Num- 

bers and  Logarithms.  .  9 

8.  Multiplication 9 

9.  Division 9 

10.  Powers 9 

11.  Roots 10 

12.  Application  to  Slide  Rules  10 

III.  MECHANICAL     CONSTRUC- 
TION. 

13.  Mechanical  Principles.  .  .  10 

14.  Body 11 

15.  Slide 11 

16.  Runner 12 

17.  Length  of  Rules 12 

18.  Graduation  of  Scales 12 

19.  Care  of  Rules 12 

20.  Method  of  Operation.  ...  12 

IV.  NOTATION  OF  SCALES. 

21.  Designation  of  Scales ....  13 

22.  Relation  of  Divisions ....  13 

23.  Scales  C  and  D 14 

24.  Scales  A  and  B 14 

25.  Reading  Scales  C  and  D  ..  15 

26.  Reading  Scales  A  and  B. .  15 

27.  Test  of  Accuracy  of  Divi- 

sion    15 

V.  MULTIPLICATION. 

28.  Two  Factors 16 

29.  Alternative  Method 17 

30.  Continued  Multiplication  17 

31.  Constant  Multiplier 17 

32.  Decimal  Point 18 

33.  Examples 18 

VI.  DIVISION. 

34.  Two  Numbers 19 

35.  Alternative  Method 19 

36.  Continued  Division 19 

37.  Reciprocals 20 

38.  Constant  Dividend 20 

39.  Constant  Divisor 20 

40.  Decimal     and     Common 

Fractions 20 

41.  Decimal  Point 21 

42.  Examples 21 

VII.  PROPORTION. 

43.  Definition 21 

44.  Direct  Proportion 21 


PAGE 

45.  Inverse  Proportion 22 

a 

46.  Solution  of  —  X  x 22 

b 
aXbXc 

47.  Solution  of  23 

dXeXf 

48.  Decimal  Point 23 

VIH.  POWERS  AND  ROOTS. 

49.  Relation    of    Upper    and 

Lower  Scales 23 

50.  Multiplication,      Division 

and  Proportion  with  A 

and  B 24 

51.  Squares 24 

52.  Square  Roots 24 

53.  Cubes 25 

54.  Cube  Roots 25 

55.  Higher  Powers 25  ' 

56.  Fractional  Powers 25 

57.  Powers  with  Proportion- 

al Dividers 26 

IX.  INVERTED  SLIDE. 

58.  Reciprocals 26 

59.  Multiplication   and  Divi- 

sion    26 

60.  Inverse  Proportion 26 

61.  Cube  Roots 27 

X.  COMBINED  SETTINGS. 

62.  List  of  Settings 27 

XI.  SCALE  OF  SINES. 

63.  Notation  of  Scale 28 

64.  Natural  Sines 28 

65.  Sines  of  Angles  40°  to  90°  29 

66.  Sines  of  Small  Angles 29 

67.  Multiplication   and   Divi- 

sion of  Sines 29 

68.  Natural  Cosines 30 

69.  Natural  Secants 30 

70.  Natural  Cosecants 30 

71.  Natural  Versed  Sines  and 

Coversed  Sines 30 

XII.  SCALE  OF  TANGENTS. 

72.  Notation  of  Scale 31 

73.  Natural  Tangents 31 

74.  Tangents   of   Angles   45° 

to  90° 31 

75.  Multiplication  and  Divi- 

sion of  Tangents 31 

76.  Natural  Cotangents 31 

77.  Solution  of  Triangles 32 

XIII.  SCALE  OF  LOGARITHMS. 

78.  General 32 

79.  Characteristic 32 

80.  Powers  and  Roots 32 


tGioonoo 


CONTENTS-PART    II. 
Multiplex    Slide    Rule. 


PAGE 

I.  INTRODUCTION. 

1.  Application 37 

2.  Accuracy 37 

3.  Saving  in  Time 37 

4.  Mechanical  Advantages. .  38 

5.  Note 38 

H.  CONSTRUCTION. 

6.  General 38 

7.  Slide 38 

8.  Reciprocal  Scale 39 

9.  Cube  Scale 39 

10.  Reading  Scale  Br 39 

11.  Reading  Scale  E 40 

HI.  MULTIPLICATION. 

12.  Mechanical  Principles .  ..  40 

13.  Two  Factors 40 

14.  Three  Factors 40 

15.  Constant  Product 40 

16.  Proportion 41 

17.  Decimal  Point 41 

18.  Examples 41 


PAGE 

IV.  DIVISION. 

19.  Mechanical  Principles ...  42 

20.  Constant  Dividend 42 

21.  Reciprocals 42 

22.  Continued  Division 42 

23.  Decimal  Point 42 

24.  Examples 43 

V.    POWERS  AND  ROOTS. 

25.  Genera] 43 

1 

26.  Solution  of  — 43 

a2 
1 

27.  Solution  of  — 43 

v^a 

28.  Cubes 43 

29.  Cube  Roots 44 

30.  Three-halves  Powers. ...  44 

31.  Two-thirds  Powers 45 

32.  Other  Powers  and  Roots-  45 

VI.  SETTINGS  FOR  THE   MUL- 
TIPLEX SLIDE  RULE. 

33.  List  of  Settings 45 


PART  I. 
THE   MANNHEIM   SLIDE   RULE 


Part  I — Mannheim  Slide  Rule 


I.     INTRODUCTION. 

1.  APPLICATION. — A  slide  rule  is  an  instrument  having 
fixed  and  movable  parts  and  employing  logarithmic  scales  by 
means  of  which  arithmetical,  algebraic  and  trigonometrica 
calculations  may  be  performed  mechanically.  Thus  the  instru- 
ment is  applicable  to  nearly  all  forms  of  calculation,  and  owing 
to  these  properties  it  is  becoming  recognized  with  increased 
rapidity  in  almost  all  branches  of  commerce  and  engineer- 
ing. Although  slide  rules  have  been  employed  by  professional 
men  for  a  comparatively  short  time,  yet  their  service  in  many 
directions  is  so  clearly  marked  that  their  use  is  now  demanded 
in  many  places.  To  the  active  engineer  and  student  the  slide 
rule  is  invaluable,  while  the  merchant,  manufacturer,  account- 
ant, statistician  and  almost  everyone  connected  in  any  way 
in  a  business  undertaking  will  find  in  it  an  instrument  of  ma- 
terial service  and  accuracy.  In  the  following  text  an  attempt 
has  been  made  to  include  all  that  is  useful  in  a  slide  rule  tc 
the  engineer  and  student.  Other  readers  will  find  it  necessary 
to  understand  only  those  parts  of  the  book  dealing  with  the 
ordinary  examples  of  multiplication,  division  and  proportion 

2.  QUALIFICATIONS.— Let  the  reader  clearly  understand 
at  the  outset  that  the  principles  which  underlie  the  theory  and 
practical  application  of  the  slide  rule  are  so  few  and  so  simple 
that  its  proficient  use  may  be  easily  understood  and  mastered 
by  almost  everyone.  The  theory  of  the  slide  rule  lies  in  the 
elementary  principles  of  logarithms  and  its  practical  applica- 
tion is  reduced  to  the  ability  required  for  reading  graduated 
scales. 

3.  ACCURACY. — The  degree  of  exactness  to  which  results 
may  be  found  depends  upon  the  skill  of  the  operator,  the 
length  of  the  scales  and  the  accuracy  of  their  division.  Pro- 
ficiency in  setting  and  reading  comes  naturally  to  the  operator 
together  with  confidence  and  certainty  in  the  results.  The 
principles  upon  which  the  slide  rule  is  based  are  infallible,  but 
a  slight  error  enters  into  computations  involving  numbers  oi 
many  figures,  due  to  the  fact  that  interpolation  is  then  neces- 
sary in  setting  and  reading.  Roughly  speaking,  the  accuracy 
obtainable  with  the  common  ten-inch  slide  rule  is  equivalent 


S  Mannheim  Slide  Rule 

to  that  of  a  three-place  logarithmic  table,  while  a  rule  twenty- 
inches  long  will  generally  add,'  another  figure  to  the  results. 
This  degree  of  precision  i$  pim^ieiat  for  almost  all  engineering 
calculations  and  is  of  material' value,  at  least  as  a  certain  check, 
for  the  most;  part  off  a»l'  otfyer/  computations  of  an  ordinary 
nature.  With  the  full-leiigtl)  soajes  o'f  the  ten-inch  rule,  results 
should  always  be  accurate  within  three-tenths  of  one  per  cent., 
while  a  little  experience  and  care  will  better  this  to  two-tenths 
and  less  even  in  rapid  working.  For  the  upper  scales  of  the 
ten-inch  rule  the  error  may  amount  to  one-third  per  cent. , 
while  with  a  twenty-inch  rule  it  is  proportionately  decreased. 

4.  SAVING  IN  TIME  AND  LABOR.— The  fact  that  the  slide 
rule  will  just  as  readily  solve  problems  involving  any  number 
of  factors  with  any  combination  of  figures  in  each,  results  in 
a  time  and  labor  saving  device  of  much  importance.  Further- 
more, it  is  almost  as  easy  to  multiply,  divide,  extract  the  root 
or  raise  to  a  power,  numbers  of  many  figures  as  it  is  to  per- 
form the  same  operations  on  those  of  the  simplest  kind.  Let 
the  reader  but  understand  the  following  text  and  then  with  a 
little  practice  and  concentration  a  considerable  amount  of 
time,  labor  and  mental  strain  will  be  eliminated  from  his  daily 
calculations. 

II.     THEORY  OF  LOGARITHMS. 

5.  DEFINITION.— To  understand  the  theory  and  action  of 
a  slide  rule  it  is  necessary  to  be  familiar  with  the  elementary 
principles  of  logarithms.  These  principles  are  primarily  based 
upon  the  fact  that  every  number  is  equal  to  some  power  of 
every  other  number,  the  exponent  or  index  of  which  power  is 
either  greater  or  less  than  one.  For  example,  any  number,  as 
49,  is  equal  to  any  number, as  10,  raised  to  a  certain  power,  the 
exponent  of  the  power  in  this  case  being  approximately  1.69. 
If  10  be  chosen  as  the  fixed  number  which  is  to  be  raised  to  a 
power  to  produce  any  other  number,  then  10  becomes  the  base 
of  this  system,  the  exponent  in  any  case  being  the  logarithm. 
In  general  the  logarithm  of  any  number  is  the  exponent  of  the 
power  to  which  the  base  of  the  system  must  be  raised  to  pro- 
duce that  number.  Thus  2  is  the  logarithm  of  100  to  the  base 
10  since  102  =  100.  The  whole  part  of  the  logarithm  which 
precedes  the  decimal  point  is  called  the  characteristic,  while 
the  decimal  part  following  it  is  the  mantissa.  In  the  loga- 
rithm 1.69,  1  is  the  characteristic  and  .69  is  the  mantissa. 

6.  COMMON  LOGARITHMS.— The  system  of  logarithms 
having  10  for  its  base  is  called  the  common  system.  For  this 
system  the  characteristic  simply  determines  the  position  of  the 
decimal  point  in  the  number  corresponding  to  the  logarithm, 
while  the  mantissa  of  the  logarithm  is  identical  for  the  same 
series  of  figures  no  matter  where  the  decimal  point  in  the  num- 
ber is  placed.     Therefore  if  the  position  of  the  decimal  point 


Theory  of  Logarithms  9 

be  neglected  only  the  mantissa  of  the  common  logarithm  need 
be  considered.  Herein  lies  the  peculiar  advantage  of  the 
common  system  of  logarithms,  and  for  this  reason  it  is  always 
applied  to  the  slide  rule.  In  the  following  text  wherever 
logarithms  are  mentioned  the  common  system  will  be  un- 
derstood. 

7.  RELATION  BETWEEN  NUMBERS  AND  LOGARITHMS. 

— As  stated,  the  mantissa  or  decimal  part  of  the  logarithm 
in  any  case  depends  only  upon  the  string  of  figures  comprising 
the  number.  These  mantissa?  have  been  tabulated  in  many 
books  and  will  be  found  as  given  in  Table  I. 


TABLE  I. 

Number.  .  .  , 
Logarithm.  . 

...     1 
..    0 

2 

.301 

3         4         5 
.477   .602    .699 

6 

.778 

7 
.845 

8 
.903 

9       10 
.954  1.000 

The  logarithm  of  any  number  composed  of  2  and  as  many 
zeros  as  you  please  will  always  have  .301  for  its  mantissa,  but 
its  characteristic  will  depend  upon  the  number  of  figures  before 
the  decimal  point;  and  similarly  for  any  other  of  the  above 
numbers. 

8.  MULTIPLICATION.— It  will  be  observed  from  Table  I 
that  the  sum  of  any  two  logarithms  is  the  logarithm  of  the 
product  of  the  two  corresponding  numbers.  For  example,  the 
sum  of  the  logarithms  of  the  numbers  2  and  3  is  .301  +  .477  or 
.778,  which  is  the  logarithm  of  6  or  2  X  3.  Similarly  the  sum 
of  the  logarithms  of  3,  5  and  6  is  1.954,  of  which  the  mantissa 
.954  is  the  logarithm  of  9,  the  characteristic  1  indicating  that 
there  are  two  figures  in  the  result  before  the  decimal  point. 
Hence  their  product  is  90  or  3X5X6.  This  same  relation  will 
be  observed  between  any  two  or  more  numbers  and  their  loga- 
rithms. Therefore  by  adding  the  logarithms  of  any  two  or 
more  numbers  the  logarithm  of  their  product  is  obtained, 
from  which  the  product  itself  may  be  easily  found. 

9.  DIVISION.— From  Table  I  it  will  also  be  apparent  that 
the  difference  between  any  two  logarithms  is  the  logarithm  of 
the  quotient  of  the  corresponding  numbers.  The  difference 
between  the  logarithms  of  8  and  4  is  .903  —  .602  or  .301,  which 
is  seen  to  be  the  logarithm  of  2  or  8  ^4;  and  similarly  for  any 
two  or  more  numbers  and  their  logarithms.  The  general  rule 
then  follows  that  by  subtracting  the  logarithm  of  one  or  more 
numbers  from  the  logarithm  of  any  number,  there  results  the 
logarithm  of  the  quotient,  from  which  the  quotient  itself  may 
be  readily  obtained. 

10.  POWERS.— If  the  logarithm  of  any  number  in  Table  I 
be  multiplied  by  2,  the  resulting  logarithm  corresponds  to  its 
second  power  or  square.  Thus  the  logarithm  of  3  multiplied 
by  2  is  .477  X  2  or  .954,  which  is  the  logarithm  of  9  or  32.  Also 
if  the  logarithm  of  2  be  multiplied  by  3  there  results  .903,  the 
logarithm  of  8  or  23.     Since  this  relation  is  true  for  any  num- 


10  Mannheim  Slide  Rule 

ber  to  any  power,  the  rule  follows  that  the  logarithm  of  a  num- 
ber multiplied  by  any  factor  gives  as  a  product  the  logarithm 
of  that  power  of  the  number  of  which  the  factor  is  the  index. 
The  power  itself  may  then  be  observed. 

11.  ROOTS. — Again  referring  to  Table  I,  it  will  be  seen  that 
by  dividing  the  logarithm  of  any  number  by  2,  the  logarithm 
corresponding  to  its  square  root  is  obtained.  Thus  the  loga- 
rithm of  9  divided  by  2  is  .954  -f  2  or  .477,  which  is  the  loga- 
rithm of  3  or  V9.  Also  .903  -8- 3  equals  .301,  which  is  the  loga- 
rithm of  2  or  the  cube  root  of  8.  Since  the  principle  is  similar 
for  the  fourth,  fifth  or  any  other  whole  or  decimal  root  of  any 
number,  the  rule  may  be  stated  that  by  dividing  the  logarithm 
of  any  number  there  is  obtained  the  logarithm  of  its  root  of 
which  the  divisor  is  the  index.  The  root  itself  may  then  be 
found. 

12.  APPLICATION  TO  SLIDE  RULES— It  has  been  seen 
that  the  processes  of  multiplication  and  division  reduce  to  the 
addition  and  subtraction  of  logarithms.  Hence  if  an  instru- 
ment is  produced  which  is  capable  of  mechanically  adding  and 
subtracting  these  logarithms,  it  may  perform  all  computations 
of  multiplication  and  division.  Likewise  the  second  or  third 
powers  and  roots  may  be  mechanically  determined  if  the  in- 
strument is  capable  of  multiplying  and  dividing  the  logarithms 
by  2  or  3. 

The  slide  rule  accomplishes  these  results  by  applying  the 
principles  of  logarithms .  Instead  of  tabulating  the  logarithms , 
the  slide  rule  carries  them  in  the  form  of  scales  or  graduated 
lengths,  each  unit  length  representing  equal  parts  of  the  loga- 
rithmic table.  Thus  if  the  logarithm  of  10  be  chosen  as  the 
unit,  then  the  logarithm  of  2  or  .301  will  be  represented  by 
.301  of  that  unit;  3  by  .477  of  the  unit;  4  by  .602;  and  so  on, 
as  can  be  determined  from  Table  I.  The  numbers  between 
1  and  2,  2  and  3,  3  and  4,  etc.,  are  represented  on  this  loga- 
rithmic scale  by  intermediate  divisions,  the  entire  scale  being 
graduated  as  closely  as  is  convenient  for  reading.  It  will  be 
observed  however,  that  the  values  of  the  logarithms  them- 
selves are  not  shown  on  the  scales,  but  instead  will  be  found 

301 
the  numbers  corresponding  to  those  logarithms.     At  tk™  th 

part   along    the    scale  on    the   slide  rule  will  be  found  2, 

845 
not  .301;  similarly  at  the    n  n  th  part  is  found  7;  the  result 

of  which  is  that  the  process  of  finding  the  numbers  correspond- 
ing to  the  logarithms  and  vice  versa  is  entirely  eliminated 
from  all  calculations. 

III.     MECHANICAL  CONSTRUCTION. 

13.  MECHANICAL  PRINCIPLES.— If  two  ordinary  scales 
be  drawn  so  that  their  divisions  are  equal  throughout,  the 


Mechanical  Construction 


11 


sum  of  any  number  of  units  of  one  scale  and  any  number  of 
the  other  may  be  determined  mechanically.     In  Fig.  1,  any 


0            12. 

SCf*  LE  C 
4             5              6 

!            8            9          U 

0           1?. 

4             5             6             7             I 
SC  ft L €  D 

9           10 

FIG.   1.  ADDITION  AND  SUBTRACTION  WITH  LINEAR  SCALES. 

number  on  scale  C  is  in  contact  with  the  sum  of  that  number 
and  3  on  scale  D.  Thus  3  of  C  and  6  of  D  are  opposite  one 
another,  and  similarly  for  4  and  7,  5  and  8,  and  so  on  for  the 
full  range  of  the  scales.  Conversely  for  subtraction,  any  two 
numbers  whose  difference  is  3  will  be  found  in  contact  on  C 
and  D.  Furthermore,  if  scale  C  be  turned  upside  down  with 
respect  to  D,  it  is  evident  that  the  above  conditions  for  addi- 
tion and  subtraction  are  reversed. 

These  principles  will  be  obvious  to  the  reader  and  embody 
everything  that  is  necessary  to  understand  the  mechanical 
principles  of  the  slide  rule.  It  will  be  remembered,  however, 
that  in  adding  the  logarithms  of  numbers,  the  logarithm  of 
their  product  results;  and  in  subtracting  them  the  difference 
is  the  logarithm  of  the  quotient.  Furthermore,  the  gradua- 
tions on  the  scales  of  the  slide  rule  are  not  equally  spaced  since 
the  numbers  are  noted  and  not  their  logarithms.  The  features 
of  mechanical  construction  of  the  slide  rule  adopted  by  the 
Eugene  Dietzgen  Co.  so  as  to  best  accomplish  these  results  will 
be  found  in  the  following  paragraphs. 

14.  BODY. — The  slide  rule  is  composed  of  three  separate 
parts,  the  body,  slide  and  runner,  the  first  two  being  made  of 
well-seasoned  and  especially  selected  wood.  The  body  is  the 
fixed  part,  consisting  of  a  base  upon  which  are  rigidly  mounted 
two  graduated  rules  made  exactly  parallel  to  each  other  and 
separated  by  an  opening  for  the  reception  of  the  slide.  The 
rules  are  faced  with  celluloid  and  have  logarithmic  scales  en- 
graved upon  them.  The  under  side  of  the  base  carries  a  table 
of  constants  for  reference,  while  along  the  opposite  sides  are 
the  ordinary  scales  of  inches  and  centimeters  for  linear  meas- 
urements. 

15.  SLIDE. — This  is  a  comparatively  thin  slip  of  wood  faced 
on  the  top  and  bottom  with  celluloid  upon  which  logarithmic 
scales  are  engraved.  Along  each  edge  of  the  slide  is  an  ex- 
tended tongue  which  accurately  fits  corresponding  grooves  in 
the  body  of  the  rule.  This  construction  allows  the  slide  to 
move  lengthwise  within  the  base  in  either  normal  or  inverted 
position  and  with  either  face  uppermost.  The  scales  of  the 
fixed  rules  and  slide  should  lie  absolutely  in  one  plane  and  no 
appreciable  opening  should  appear  at  the  contact  edges  of  the 


12  Mannheim  Slide  Rule 

scales  for  any  position  of  the  slide.  Furthermore,  the  slide 
should  move  just  freely  enough  so  as  to  neither  bind  nor  stick 
and  still  be  secure  in  every  position. 

16.  RUNNER. — The  runner  is  formed  of  a  small  aluminum 
frame  enclosing  a  piece  of  glass.  On  the  underside  and  along 
the  center  of  the  glass,  lying  almost  in  contact  with  the  scales, 
is  a  fine  hair-line.  The  runner  slides  in  grooves  on  the  sides 
of  the  base,  while  a  spring  on  one  side  permits  free  movement 
along  the  length  of  the  rule,  but  always  holds  the  hair-line  truly 
at  right  angles  to  all  scales.  The  hair-line  should  be  about  as 
heavy  as  the  scale  graduations,  and  uniform  throughout  its 
length.  The  runner  is  used  for  settings  and  for  referring  from 
one  scale  to  the  other;  it  also  eliminates  the  necessity  of  read- 
ing intermediate  results  in  a  continued  series  of  calculations. 

17.  LENGTH  OF  RULES.— Slide  rules  are  made  by  us  in 
three  standard  sizes,  the  full  length  of  the  logarithmic  scales 
being  12.5,  25  and  50  centimeters  respectively.  The  rule  al- 
most invariably  used,  however,  has  the  25-centimeter  scale  and 
is  popularly  known  as  the  ten-inch  slide  rule.  This  length 
best  combines  accuracy  and  convenience,  although  the  twenty- 
inch  size  is  more  accurate  and  the  five-inch  more  convenient 
to  carry  around.  The  stock  of  the  body  and  slide  project  a 
little  beyond  the  scales  in  order  to  secure  a  firm  setting  of  the 
slide  and  runner  at  either  end. 

18.  GRADUATION  OF  SCALES.— The  scales  of  our  rules 
are  engine  divided  and  engraved,  the  divisions  being  automat- 
ically spaced  by  means  of  a  logarithmic  screw.  This  process 
results  in  scales  of  highest  accuracy  and  greatest  durability. 
The  lines  are  black  on  a  clear  white  celluloid  background,  thus 
producing  a  reading  surface  of  considerable  merit. 

19.  CARE  OF  RULES.— It  is  important  that  slide  rules  be 
kept  in  places  where  excessive  or  rapid  changes  of  tempera- 
ture or  humidity  are  not  likely.  All  varieties  of  wood  and 
celluloid  are  liable  to  shrink  and  warp  under  unfavorable  con- 
ditions, and  when  such  does  occur  the  instrument  will  become 
unsatisfactory  in  operation.  Always  keep  your  rule  away 
from  radiators  and  damp  places. 

20.  METHOD  OF  OPERATION.— It  has  been  found  most 
convenient  and  accurate  to  operate  with  the  slide  rule  flat 
down  on  a  table.  The  projecting  end  of  the  slide  should  be 
held  along  both  tongues  at  the  end  of  the  base,  between  the 
thumb  and  index  finger.  The  index  finger  of  the  other  hand 
should  be  kept  between  the  fixed  rules  and  against  the  end  of 
the  slide,  with  the  thumb  and  middle  finger  of  that  hand  along 
the  sides  of  the  base.  The  slide  may  then  be  easily  and  grad- 
ually moved  between  the  fingers  of  the  first  hand,  which  are 
capable  at  any  time  of  reducing,  stopping  or  reversing  the 
motion  of  the  slide.  The  operator  will  appreciate  this  method 
after  a  little  practice  with  the  rule. 


Notation  of  Scales 


13 


IV.     NOTATION  OF  SCALES. 


21.  DESIGNATION  OF 

SCALES.— There  are  six 
complete  logarithmic 
scales  on  both  the  fixed 
rules  and  the  upper  face 
of  the  slide.  The  upper 
fixed  rule  carries  two 
scales  which  are  identical 
with  the  two  engraved 
along  the  upper  edge  of 
the  slide.  Along  the 
lower  edge  of  the  slide 
and  the  lower  fixed  rule 
are  two  complete  scales. 
The  upper  scales  are 
usually  designated  A 
and  B,  while  the  lower 
ones  are  marked  C  and 
D,  as  shown  in  Fig.  2. 
Scale  A  includes  the  two 
complete  logarithmic 
scales  A'  and  A" \  while 
B  includes  B'  and  B". 
The  initial  graduations 
at  the  left-hand  end  of 
the  scales  A,  B,  C  and 
D  are  each  termed  the 
left  index  of  its  respec- 
tive scale,  while  the  cor- 
responding lines  at  the 
right-hand  end  are  the 
right  indices.  The 
middle  graduations  of 
scales  A  and  B  are 
called  their  center  in- 
dices. 

22.  RELATION  OF 
DIVISIONS.— Each  log- 
arithmic scale  is  laid  off 
from  its  left  index, 
which  being  the  start- 
ing-point is  marked  1, 
since  the  logarithm  of  1 
is  0.  (  See  Table  I.  ) 
Each  other  number  is 
represented  by  the  same 
part  of  the  chosen  unit 
length  as  its  logarithm 
bears  to  the^logarithm 


i*«»H 


Odd 


LEFT 
fNOfX 


tcmjH 

flNDCX 

I 


fcplGHT 
INDEX 


T  J" 


f 


i 
i 
i 

i! 

o 
r 


r-  ~ 


I 
I 


1 1.. 


14 


Mannheim  Slide  Rule 


of  the  chosen  unit.  Then  since  10  is  taken  as  the  unit,  the  num- 
ber 2  will  be  found  at  the  distance  .301  from  the  left  index  of 
the  scale;  3  is  located  at  .477;  4  at  .602;  and  similarly  for  every 
other  number.  In  this  way  the  complete  logarithmic  scales 
are  laid  out,  it  being  observed  that  each  number  is  repre- 
sented by  a  certain  length  in  terms  of  the  length  of  the  scale. 

From  what  has  been  said  in  reference  to  the  common  system 
of  logarithms  it  will  be  apparent  that  the  left  index  of  any 
scale  may  represent  1, 10, 100,  .1,  .01,  .001,  etc.,  in  which  cases 
the  division  marked  2  will  represent  2,  20,  200,  .2,  .02,  .002, 
etc.,  respectively;  and  similarly  for  all  other  numbers.  Simply 
remember  that  the  divisions  marked  2,  3,  4,  etc.,  are  respec- 
tively 2,  3,  4,  etc.,  times  the  chosen  value  of  their  left  index. 

23.  SCALES  C  AND  D— These  scales  are  similar  to  each 
other  in  every  respect.  In  order  that  they  may  be  graduated 
as  minutely  at  all  parts  as  is  convenient  to  read,  it  is  necessary 
to  have  three  different  values  for  their  smallest  divisions. 
With  the  left-hand  index  equal  to  1,  the  values  of  all  divisions 
on  scales  C  and  D  are  shown  in  Table  II  for  the  three  sizes  of 
rules. 

TABLE  II. 


FIVE  AND  TEN-INCH 
RULES. 

TWENTY-INCH 
RULE. 

1to2 

2  to  4  4  to  10 

1  to  2  12  to  5 

5  to  10 

Value  of  each  main  division 
Value  of  each  minor   divi- 

1.00 

.10 
.01 
100 

1.00 

.10 
.02 
100 

1.00 

.10 
.05 
120 

1.000    1.00 

. 100      .  10 
.005       .01 
2001     300 

1.00 
.10 

Value  of  smallest  division. . 
Total  number  of  divisions. . 

.02 
250 

In  general  the  C  and  D  scales  are  used  for  the  simple  calcu- 
lations where  accuracy  is  desired,  this  feature  resulting  from 
the  greater  length  and  subdivsion  of  those  scales. 

24.  SCALES  A  AND  B. — These  comprise  the  four  complete 
logarithmic  scales  A',  A",  B'  and  B"  on  the  upper  fixed  rule 
and  along  the  upper  edge  of  the  slide.  Each  is  exactly  one- 
half  the  length  of  that  of  the  C  or  D  scale,  the  left  indices 
of  A'  and  B'  and  the  right  indices  of  A"  and  B"  being  engraved 
accurately  in  line  with  the  left  and  right  indices  respectively 
of  the  lower  scales.  The  values  of  the  divisions  of  each  of  the 
upper  scales  are  shown  in  Table  III  for  the  five,  ten  and  twenty - 
inch  rules  with  their  left  index  equal  to  1. 


TABLE  III 

FIVE  AND  TEN-INCH 
RULES. 

twenty-inch 

RULE. 

1to2 

2  to  5 

5  to  10 

1to2 

2  to  4 

4  to  10 

Value  of  each  main  division 
Value  of  each  minor   divi- 

1.00 

.10 

.02 

50 

1.00 

.10 
.05 
60 

1.00 

!io 

50 

1.00 

.10 
.01 
100 

1.00 

.10 
.02 
100 

1.00 
.10 

Value  of  smallest  division. . 
Total  number  of  divisions. . 

.05 
120 

Notation  of  Scales 


15 


On  scales  A'  and  B'  will  be  found  a 
which  is  marked  it.  in  each  case,  and 
on  scales  A"  and  B"  is  located  .7854 
or  7T-J-4.  Those  constant  demarca- 
tions are  for  convenience  in  the  fre- 
quently recurring  calculations  of  areas 
and  diameters  of  circles.  Scales  A  and 
B  are  usually  employed  for  computa- 
tions where  rapidity  in  working  is  the 
main  consideration,  and  for  finding 
powers  and  roots. 

25.  READING  SCALES  C  AND  D.— 
For  rapid  working  with  the  slide  rule 
it  is  necessary  that  the  operator  be 
accurate  and  confident  in  his  settings 
and  readings  of  the  various  scales. 
The  C  and  D  scales  are  divided  and 
subdivided  as  shown  in  Table  II,  and 
for  further  figures  it  is  necessary  to 
estimate  the  decimal  part  of  the  small- 
est divisions  by  the  eye.  Various 
examples  of  settings  and  interpolations 
appear  in  Fig.  3,  along  different  parts 
of  the  scale  of  a  ten-inch  rule  and  with 
the  left  index  taken  equal  to  1,000. 
Besides  these,  the  reader  should  make 
the  following  and  many  other  settings 
and  readings, using  both  the  upper  and 
lower  scales:  Set  8  on  C  to  17  on  D  and 
read  2.125  on  D  at  the  right  index  of 
C;  set  4  on  C  to  53  on  D  and  at  the 
left  index  of  C  read  13.25  on  D;  set  68 
on  C  to  the  right  index  of  D  and  by 
means  of  the  runner  read  8.25  on  D  at 
561  on  C. 

26.  READING  SCALES  A  AND  B.— 

The  values  of  the  various  divisions  on 
these  scales  are  given  in  Table  III, 
and  are  seen  to  differ  from  those  of  the 
lower  scales.  Various  readings  along 
the  upper  scales  are  shown  in  Fig.  4 
(see  page  16)  for  a  ten-inch  rule,  and 
after  these  have  been  carefully  com- 
pared the  reader  should  make  many 
other  settings  and  readings  both  with 
and  without  the  runner,  until  he  is 
thoroughly  familiar  with  all  parts  of 
the  upper  and  lower  scales. 

27.  TEST  OF  ACCURACY  OF  DI- 
VISION. —  Although    our    scales    are 


graduation 

at  3.1416 

-  iOOO 

- 1030 

~ 

1061 

&3= 

ii/» 

nut 

_= 

II4J 

_E 

i  ifli 

1  loft  i 

<\T^ 

*j 

is 

-  \115 

s 

— ■■— 

— 

m 

CO 

<M    — 

W7 

8 

~ 

I1CQ 

IJJO 

g 

~ 

0 

_— t^ 

-      \333 

3 

0 

cc 

o 

ZQ00 

3 

-  -  m 

CO 

o 

5125 

t> 

r 

2190 

H 

o 

- ??6S 

o 

■■■■-  13S5 

& 

0 

U35 

o 

£== 

-     ?& 

> 

ZQ0 

mo 

1 

1800 

3 

o 

B 

-     7500 

GO 

s 

77\0 

H 

td 

737.0 

8\?S 

P 

-  ■  8330 

--- 6550 

8760 

qctjd 

OJlU 

s 

-  9175 

-9380 

9630 

t±. — = 

10000 

16 


Mannheim  Slide  Rule 


accurately  divided  the 

1000 


zm 

1770 

*— = 

-  -  3\?5 

■--  3625 

zooo 


mo 


MO 


2990 


■3350 


3900 


reader  should  make  an  absolute  test 
on  all  scales.  First  set  the  left  indices 
of  scales  C  and  D  together  and  note 
that  all  other  corresponding  indices 
and  graduations  are  in  exact  contact. 
Then  by  means  of  the  runner  see  that 
the  four  left  indices  are  exactly  in  line 
and  similarly  for  the  four  right  indices, 
which  process  will  simultaneously  check 
the  alignment  of  the  hair-line  of  the 
runner.  Now  set  2  on  scale  C  to  the 
left  index  of  D  and  note  that  all  gradu- 
ations between  2  and  4  on  scale  C 
exactly  coincide  with  the  graduations 
on  D  between  1  and  2.  Then  set  4 
on  C  to  8  on  D  and  observe  exact 
coincidence  of  divisions  from  4  to  5  on 
C  with  8  to  10  on  D.  Finally  set  5  on 
C  to  the  left  index  of  D  and  see  that 
all  divisions  between  5  and  the  right 
index  of  C  are  in  absolute  contact  with 
graduations  of  scale  D. 

To  completely  check  scales  A  and 
B,  set  the  left  index  of  B'  to  the  left 
index  of  A"  and  observe  alignment  of 
all  corresponding  graduations  through- 
out these  scales.  Then  set  2  on  B"  to 
4  on  A'  and  note  contact  for  all  divis- 
ions between  2  and  the  right  index 
of  B". 

V.    MULTIPLICATION. 

28.  TWO  FACTORS.— It  has  been 
explained  that  the  sum  of  the  loga- 
rithms of  two  or  more  numbers  is  the 
logarithm  of  the  product  of  those  num- 
bers; also  the  logarithms  of  numbers 
are  mechanically  added  by  means  of 
the  slide  and  fixed  rules.  Hence  to 
multiply  any  two  numbers,  using  scales 
C  and  D,  one  index  of  C  is  set  di- 
rectly above  either  f  ctor  on  D  and 
under  the  other  factor  on  C  is  read  their 
product  on  D.  This  statement  may  be 
expressed  in  tabular  form  as  below  for 
both  the  C  and  D  scales  and  A  and  B, 
the  latter  including  A'  or  A"  with  either 
B'  or  B". 


Set  1 


At  other  number  1 1  B 


to  one  number    I  (  read  product 


Multiplication  17 

It  will  be  observed  after  choosing  any  set  of  values  that  but 
one  of  the  two  indices  of  C  can  be  used  with  D  for  any  given 
case.  If  25  is  to  be  multiplied  by  31,  the  left  index  of  C  must 
be  set  to  either  25  or  31  in  order  that  the  other  factor  on  C 
will  be  above  scale  D;  and  if  635  is  to  be  multiplied  by  2,  the 
right  index  of  C  must  be  used  in  order  to  read  their  product 
1270  under  scale  C.  The  following  examples  should  be  solved, 
using  both  the  upper  and  lower  sets  of  scales,  and  then  checked 
by  multiplying  the  factors  out  by  hand : 

CllSetl    I  At  25  II  B  C  ||Set  1    |At  3  MB 


D  1 1  to  636  |readproductl5900ll  A  D  llto  259  Iread  product  777  II  A 

With  the  slide  projecting  on  the  right  the  logarithm  of  one 
factor  on  one  scale  is  added  to  the  logarithm  of  the  other  fac- 
tor on  the  other  scale,  but  if  the  slide  projects  on  the  left  of 
the  rule,  10  minus  the  logarithm  of  the  factor  on  the  slide  is 
subtracted  from  the  logarithm  of  the  factor  on  the  fixed  rule. 
However,  the  results  of  these  two  operations  are  exactly  equiv- 
alent, since  for  the  latter  case 

log.  a  —  (10  —  log.  b)  =  log.  a  +  log.  b  —  10, 
wherein  a  and  b  represent  any  two  numbers  while  10  is  the 
value  of  the  full  scale  length. 

29.  ALTERNATIVE  METHOD.— The  following  method  of 
multiplication  is  sometimes  convenient,  although  the  preced- 
ing form  is  usually  adopted: 

C  II     Set  one  number        I     read  product  II  B 

D II     to  1  I     At  other  number      ||  A 

In  the  above  the  logarithms  of  the  two  numbers  are  added 
together,  but  in  a  different  way  from  the  method  of  paragraph 

28. 

30.  CONTINUED  MULTIPLICATION.— The  process  of  mul- 
tiplying more  than  two  numbers  together  is  exactly  similar  to 
the  method  for  two  factors,  the  product  of  the  first  two  being 
in  turn  multiplied  by  the  third  number,  and  that  product  by 
the  fourth,  and  so  on.  In  such  cases  it  will  be  found  necessary 
to  use  the  runner  in  order  to  avoid  reading  off  the  intermedi- 
ate products.  The  following  tabular  statement  for  the  multi- 
plication of  three  factors  will  be  apparent: 

1  to  runner  At  third  number 


c 

1 
Set  1 

Runner  to  sec- 
ond number 

D 

to  first  number 

read     final     prod- 
uct 


31.  CONSTANT  MULTIPLIER.— By  setting  the  index  of  C 
to  any  number  on  D,  all  products  of  the  factor  on  D  with  all 
numbers  in  contact  on  C  may  be  read  off  directly  without  re- 
setting the  slide.  This  property  of  the  slide  rule  is  often  con- 
venientin'a  series  of  multiplications  having  a  constant  factor. 


18  Mannheim  Slide  Rule 

For  such  problems  it  is  generally  preferable  to  use  A  and  B, 
since  complete  scales  are  always  in  contact,  thus  eliminating 
the  necessity  of  shifting  the  slide,  as  may  be  required  for  the 
lower  scales. 

32.  DECIMAL  POINT.— The  string  of  figures  in  the  product 
having  been  determined,  it  then  becomes  necessary  to  locate 
the  decimal  point.  It  is  always  advisable  for  the  operator  to 
mentally  check  the  problem,  thus  locating  the  decimal  point 
at  the  same  time,  which  in  many  cases  may  be  done  by  in- 
spection. However,  rules  will  be  given  which  are  simple  and 
cover  all  cases  of  multiplication.  First  it  must  be  understood 
that  the  digits  in  any  number  greater  than  1  is  the  same  as 
the  number  of  figures  preceding  the  decimal  point,  while  for 
numbers  less  than  1,  the  digits  are  minus  and  equal  to  the 
number  of  zeros  which  directly  follow  the  decimal  point. 
Thus  2.693  has  one  digit;  149.06,  three  digits;  14836,  five  digits. 
For  decimal  numbers  .103  has  zero  digits;  .09,  minus  one  digit; 
.0000238,  minus  four  digits.  Where  there  are  more  than  two 
factors  to  be  multiplied  each  setting  is  to  be  considered  sepa- 
rately. The  following  rules  apply  only  to  the  common  method 
of  multiplication  as  outlined  in  paragraph  28,  while  for  the 
alternative  method  of  paragraph  29,  the  rules  for  the  decimal 
point  are  reversed. 

FOR  SCALES  C  AND  D  ONLY. 

WITH  THE  SLIDE  PROJECTING  ON  THE  LEFT,  the  num- 
ber of  digits  in  the  product  is  equal  to  the  sum  of  the  digits 
in  the  two  factors. 

WITH  THE  SLIDE    PROJECTING  ON  THE  RIGHT,  the 

number  of  digits  in  the  product  is  equal  to  one  less  than  the 
sum  of  the  digits  in  the  two  factors. 

In  applying  these  rules  to  a  decimal  factor,  remember  that 
subtracting  from  a  minus  number  of  digits  increases  the  num- 
ber of  negative  digits,  while  adding  to  it  has  the  opposite  effect. 
For  example,  2  subtracted  from —  3  digits  is  —  5  digits,  and  2 
digits  added  to  — 3  is — 1  digit. 

The  position  of  the  decimal  point  may  be  determined  for  all 
scales  and  methods  by  remembering  that  where  the  first  sig- 
nificant figure  of  the  product  is  less  than  the  first  significant 
figure  of  both  factors,  or  equal  to  it,  in  which  case  the  succeed- 
ing figures  are  to  be  likewise  compared,  the  number  of  digits 
in  the  product  is  equal  to  the  sum  of  the  digits  of  the  two 
factors;  and  where  it  is  greater  the  digits  in  the  product  is  one 
less  than  their  sum.  By  the  first  significant  figure  is  meant 
the  first  figure  from  the  left  other  than  zero.  The  first  signifi- 
cant figure  of  309.6  is  3,  and  of  .00298  is  2. 

33.  EXAMPLES. — The  following  examples  are  appended  so 
that  the  reader  may  become  familiar  with  the  process  of  mul- 
tiplication and  the  location  of  the  decimal  point.    The  solu- 


Division 


19 


tions  are  shown  for  the  C  and  D  scales,  but  scales  A  and  B 
should  also  be  used  by  the  reader  for  each  problem. 


EXAMPLES 

SUM 

OF 

DIGITS 

NO.  OF  TIMES 
SLIDE  PROJECTS  ON 

DIGITS 

IN 

PRODUCT 

ANSWER 

LEFT 

RIGHT 

55  X  4 . 6  .  . 

3 
0 

3 

—  1 

1 
1 

2 
1 

0 

1 

1 

2 

3 

—  1 

2 

—  3 

253 

.00039X1.41X41.5. 
4.75  X    1.28    X    83.3 

X  .0351 

.0017  X    .029  X    .111 
X  68 

.0228 

17.78 
.000372 

VI.     DIVISION. 

34.  TWO  NUMBERS.— The  slide  rule  mechanically  per- 
forms division  by  subtracting  the  logarithm  of  the  divisor 
from  that  of  the  dividend.  Using  the  C  and  D  scales,  the  pro- 
cess consists  in  setting  the  divisor  on  C  over  the  dividend  on 
D  and  under  one  index  of  C  finding  the  quotient  on  D. 


C  I 

D 


Set  divisor 


At  1 


to  dividend 


read  quotient 


I  B 

I  A 


Where  the  quotient  is  found  at  the  left  index  of  either 
C  or  B  the  logarithms  are  subtracted  directly,  but  by  using 
the  right  index,  10  minus  the  logarithm  of  the  divisor  is  added 
to  the  logarithm  of  the  dividend,  which,  however,  is  exactly 
equivalent  to  the  first  case,  since 

log.  a  +  (10  —  log.  b)  =log.  a  —  log.  b  + 10, 
wherein  a  is  the  dividend,  b  the  divisor  and  10  the  value  of 
the  unit  scale  length. 

The  reader  should  perform  the  following  divisions,  using  in 
turn  all  sets  of  scales,  and  then  check  the  answer  by  hand : 

C||Set81      I  At  1  II  B     C  ||Set26    I  At  1  II  B 


D  1 1  to  28500  I  read  quotient  352 


D  I  [to  4550  Iread  quotient  175 


35.  ALTERNATIVE  METHOD.— It  is  sometimes  conven- 
ient to  divide  in  the  following  manner,  using  either  pair  of 
scales,  as  shown: 


C  I 

D~| 


Set  divisor 


At  dividend 


to   1 


read  quotient 


I  B 

I  A 


The  principle  of  this  method  is  entirely  analogous  in  results 
to  that  of  the  preceding. 

36.  CONTINUED  DIVISION.— This  operation  consists  of  a 
series  of  divisions,  the  runner  being  used  to  avoid  reading  off 


20  Mannheim  Slide  Rule 

the  intermediate  quotients.     In  tabular  form  the  process  for 
two  divisions  is  as  follows: 


c 

Set  first 
divisor 

Runner  to  1 

Second  divisor 
to  runner 

At  1 

B 

D 

to  dividend 

read  final  quotient     I 

A 

37.  RECIPROCALS. — The  reciprocal  of  a  number  is  equal 
to  1  divided  by  the  number.  Thus  the  reciprocal  of  4  is  .25; 
of  500  is  .002;  and  of  .0625  is  16.  The  process  is  simply  one 
of  division,  and  may  be  performed  by  either  of  the  following 
methods,  using  either  set  of  scales: 

C  1 1     Set  number  I     At  1  MB 

D 1 1     to   1  I     read  reciprocal       1 1  A 


2.1 

D| 


Set  1  I     read  reciprocal       1 1  B 


to  number  |     At  1 


Although  the  method  is  seldom  used,  the  above  principles 
may  be  applied  to  division  as  follows: 

C  II     Set  dividend  I     read  quotient         ||  B 

D  1 1     to  divisor  I     At  1  1 1  A 

By  this  process  the  divisor  is  divided  by  the  dividend  and  the 
reciprocal  of  that  quotient  found,  all  at  one  setting  and  with 
any  pair  of  scales. 

38.  CONSTANT  DIVIDEND.— With  the  ordinary  position 
of  the  slide,  such  problems  are  best  solved  by  bringing  the 
runner  to  the  constant  number  on  D  and  successively  setting 
the  divisors  on  C  to  the  hair-line.  The  quotients  are  read  in 
turn  on  D  under  the  index  of  C. 

39.  CONSTANT  DIVISOR.— Much  time  is  saved  in  problems 
of  this  kind  by  multiplying  the  reciprocal  of  the  constant 
divisor  by  the  series  of  dividends.  The  entire  set  of  quotients 
may  then  be  read  off  without  shifting  the  slide.  Where  the 
accuracy  of  the  upper  scales  is  sufficient,  they  should  gener- 
ally be  used  so  that  all  numbers  will  be  in  contact  for  any 

Eosition  of  the  slide.     This  method  of  solution  is  indicated 
elow : 

C  II     Set  constant  divisor     I     At  dividends  ||  B 


D 1 1  to   1  I     read  quotients       1 1  A 

40.  DECIMAL  AND  COMMON  FRACTIONS.— Common 
fractions  may  be  converted  into  decimals  by  dividing  the 
numerator  by  the  denominator  according  to  the  ordinary 
method  of  division.  Decimals  may  be  changed  to  common 
fractions  by  setting  1  on  the  slide  to  the  decimal  and  finding 
two  numbers  in  exact  contact.     If  the  common  fraction  is  to 


Proportion 


21 


have  a  certain  denominator  or  numerator,  the  corresponding 
numerator  or  denominator  respectively,  is  then  found  in  con- 
tact with  the  given  term. 

41.  DECIMAL  POINT.— After  the  string  of  figures  in  the 
quotient  has  been  determined,  the  decimal  point  may  be  lo- 
cated for  the  method  of  paragraph  34,  by  tne  following: 

FOR  SCALES  C  AND  D  ONLY. 

WITH  THE  SLIDE   PROJECTING   ON  THE   LEFT,  the 

number  of  digits  in  the  quotient  is  equal  to  the  digits  of  the 
dividend  less  the  digits  of  the  divisor. 

WITH  THE  SLIDE  PROJECTING  ON  THE  RIGHT,  the 

number  of  digits  in  the  quotient  is  equal  to  one  more  than  the 
digits  of  the  dividend  less  the  digits  of  the  divisor. 

The  decimal  point  may  be  located  in  a  quotient  for 
any  method  and  either  set  of  scales  from  the  fact  that  where 
the  first  significant  figure  of  the  divisor  is  greater  than  that  of 
the  dividend,  the  number  of  digits  in  the  quotient  is  equal  to 
the  digits  of  the  dividend  less  those  of  the  divisor.  If  the 
first  significant  figure  of  the  divisor  is  less  than  that  of  the 
dividend,  then  one  digit  must  be  added  to  this  difference. 
Where  the  first  significant  figures  are  the  same  the  following 
figures  must  be  compared,  as  in  the  case  of  multiplication. 

42.  EXAMPLES.— The  reader  should  solve  the  following 
problems,  using  both  sets  of  scales  in  turn: 


DIVI- 
DEND 

FIRST 
DIVI- 
SOR 

SEC- 
OND 
DIVI- 
SOR 

THIRD 
DIVI- 
SOR 

DIF- 
FER- 
ENCE 

OF 
DIGITS 

NO.  OF  TIMES 

SLIDE 
PROJECTS  ON 

DIGITS 

IN 

QUO- 

ANSWER 

LEFT 

RIGHT 

546 

182.9 
.0458 
.387 

32.9 
11.11 
.1746 
.938 

.0042 

7.21 
4.94 

l!246 

.0333 

1 

3 

—3 

0 

0 
1 

1 

2 

1 
1 

2 

1 

2 

4 

— 1 

1 

16.6 
3920 
.0292 
2.51 

VII.     PROPORTION. 

43  DEFINITION. — Proportion  is  an  equality  of  ratios. 
The  statement  that  3  is  to  6  as  4  is  to  8  is  a  proportion  in 
which  3  bears  the  same  relation  to  6  as  4  does  to  8.  The  so- 
lutions of  problems  in  proportion  for  any  of  the  unknown 
quantities  are  examples  in  combined  multiplication  and  divi- 
sion, and  are  most  conveniently  and  readily  solved  with  the 
slide  rule. 

44.  DIRECT  PROPORTION.— The  general  form  is 

1st  term:  2d  term::  3d  term:  4th  term. 

From  well-known  principles  the  product  of  the  two  outer 
terms,  1st  and  4th,  equals  the  product  of  the  two  inner  terms, 


22  Mannheim  Slide  Rule 

2d  and  3d  In  examples  of  this  kind  three  of  the  quantities 
are  given  and  the  remaining  term  is  determined  from  the  fol- 
lowing : 

CM      1st  term  I     3d  term  1 1  B 


D  1 1     2d  term  |     4th  term         1 1  A 

If  one  of  the  terms  equals  1,  then  the  process  reduces  to 
simple  multiplication  or  division  of  the  other  two  numbers. 

By  setting  two  terms  together  as  indicated  above,  it  will  be 
noted  that  all  other  numbers  which  bear  the  same  relation 
are  in  contact.  As  an  example  consider  the  following:  At 
the  rate  of  60  miles  in  2  hours,  how  far  will  a  train  travel  in 
15  hours?  25  hours?  30  hours? 

C||Set2hours      I At  15  hours  I At  25  hours         I  At  30  hours         II  B 

D  I J  to  60  miles       I  read  450  miles       I  read  750  miles     |read  900  miles     |l  A 

It  will  be  observed  that  60  is  first  divided  by  2  and  the 

quotient  found  at  one  index  of  the  slide,  which  is  the  required 

setting  for  the  multiplication  of  this  quotient  by  any  number 

within  the  range  of  contact.     In  a  similar  manner  any  prob- 

a.  X  b 
lem  in  the  form  of may  be  solved  with  one  setting  of  the 

slide. 

45.  INVERSE  PROPORTION.— Where  more  requires  less 
or  less  requires  more,  there  exists  an  inverse  form  of  propor- 
tion. With  the  ordinary  position  of  the  slide,  such  problems 
are  solved  similarly  to  examples  in  direct  proportion,  provided 
the  problem  is  stated  inversely  so  that  the  product  of  the 
outer  terms  is  equal  to  the  product  of  the  inner  terms.  For 
instance,  assume  that  8  men  perform  a  piece  of  work  in  3  days, 
how  long  will  it  take  6  men  working  at  the  same  rate  to  do 
the  same  work?  This  is  a  case  of  inverse  proportion  in  which 
having  less  men  requires  more  time;  or  in  inverse  form 

C  II     Set  8  men       I     find  4  days    I  IB 


D  1 1     to  6  men         |     At  3  days       1 1  A 
46.   SOLUTION  OF  -£-  X  x.—  Problems  of  this  kind  wherein  a, 

D 

b  and  x  represent  any  numbers  whatsoever,  are  cases  of  pro- 
portion, and  may  be  solved  in  a  single  setting  by  the  follow- 
ing method: 

C  ||     Set  b  |     At  x  MB 


read  answer    jj  A 


If  in  a  set  of  calculations  a  and  b  are  each  constant  num- 
bers and  x  has  a  series^  of  values,  this  setting  will  be  found 
convenient,  especially  if  scales  A  and  B  are  used. 


Powers  and  Roots  23 

In  using  a  constant  multiplier,  as  is  given  in  the  conversion 
ratios  to  follow  and  on  the  back  of  the  rule,  a  simple  and 
equivalent  ratio  is  noted,  rather  than  the  multiplier  itself 
which  is  usually  a  long  decimal.  For  example,  instead  of  stat- 
ing that  the  diameter  of  a  circle  multiplied  by  3.1416  equals 
its  circumference,  the  relation  between  diameter  and  circum- 
ference is  given  as  226  :  710,  since  the  quotient  of  these  num- 
bers very  closely  equals  3.1416,  and  since  they  are  points  of 
graduation  on  scales  C  and  D.  The  determination  of  any  cir- 
cumference from  its  diameter,  or  vice  versa,  is  then  as  follows : 

C  II     Set  226  |     At  diameter  II  B 


D||     to  710  I     read  circumference   1 1  A 

47.  SOLUTION  OF    **b*°  .—Problems  in  this  form  are 
dXeXf 

solved  by  a  series  of  multiplications  and  divisions.  However, 
instead  of  multiplying  the  factors  of  the  numerator  together 
and  then  dividing  in  turn  by  the  quantities  in  the  denomin- 
ator, greater  rapidity  is  usually  obtained  by  the  following 
method : 


c 

1     Setd 

Runner 
tob 

e  to  run-  1  Runner 
ner            to  c 

f  to  run- 
ner 

At  1 

15 

I) 

1     to  a 

1 

read  answer 

A 

48.  DECIMAL  POINT.— The  decimal  points  in  the  preced- 
ing problems  are  located  by  the  principles  for  multiplication 
and  division,  each  operation  being  considered  separately.  The 
following  method  of  keeping  record  in  multiplication  and  di- 
vision is  recommended  for  long  problems :  For  each  time  that 
an  extra  digit  is  to  be  added  the  sign  |  is  noted,  and  for  each 
time  an  extra  digit  is  to  be  subtracted  the  sign  —  is  set  down 
the  two  opposite  signs  being  allowed  to  cancel  each  other  as 

t  +u         -n       t    ^         ui  .042X36.9X147 

far  as  they  will.      In  the  problem  3^^00186X232 '   Per" 

formed  on  the  C  and  D  scales,  the  slide  projects  on  the  right 
twice  in  multiplying  and  three  times  in  dividing,  giving  the 
record  +  +  | ,  which  indicates  that  the  answer  contains  one  more 
than  the  digits  of  the  numerator  less  those  of  the  denominator. 
Hence  the  final  answer  has  ( — 1+2+3) — (2 — 2  +  3)  +  l  or  2 
digits,  and  equals  16.2.  * 

VIII.     POWERS  AND  ROOTS. 

49.  RELATION  OF  UPPER  AND  LOWER  SCALES.— Each 
of  the  upper  scales  is  exactly  one-half  the  length  of  the  lower 
scales,  and  with  its  corresponding  indices  accurately  in  line. 
Hence  any  logarithm  on  the  D  scale  multiplied  by  2  equals 
the  logarithm  directly  above  it  on  A;  and  similarly  for  scales 
C  and  B.  Therefore,  since  multiplying  or  dividing  the  logarithm 


24  Mannheim  Slide  Rule 

of  a  number  by  2,  gives  the  logarithm  of  the  second  power  or 
root  respectively,  of  the  number,  the  slide  rule  gives  a  direct 
means  of  determining  squares  and  square  roots  of  all  numbers. 

50.  MULTIPLICATION,  DIVISION  AND  PROPORTION 
WITH  A  AND  B. — The  upper  scales  may  be  used  for  all  such 
examples  according  to  the  methods  outlined  under  like  opera- 
tions for  the  lower  scales.  In  fact,  the  preceding  text  has 
been  made  general  so  as  to  apply  to  either  set  of  scales.  Due 
to  the  fact  that  all  numbers  are  always  in  contact,  the  upper 
scales  should  be  used  for  such  examples  where  rapid  working 
is  desired  and  the  greater  error  due  to  decreased  length  and 
subdivision  of  these  scales  is  permissible. 

51.  SQUARES. — The  squares  of  all  numbers  on  D  will  be 
found  directly  above  on  A,  and  similarly  for  scales  C  and  B. 


read  square  1 1  B 


Over  number  ||  C 


Either  the  runner  or  the  index  line  may  be  used  in  referring 
from  D  to  A,  the  latter  giving  more  reliable  results.  The 
square  of  any  number  may  also  be  found  by  multiplying  the 
number  by  itself,  preferably  using  the  lower  scales  for  the 
purpose.  By  any  of  these  methods  the  square  of  2  is  seen  to 
be  4;  5 squared  is  25;  lf.22  =296;  and  .08312  =.0069. 

If  a  square  is  read  on  A'  or  B',  the  number  of  its  digits  is 
one  less  than  twice  the  digits  of  the  given  number;  and  if  read 
on  A"  or  B",  it  has  simply  twice  the  digits  of  the  number. 

52.  SQUARE  ROOTS.— The  method  of  finding  square  roots 
is  exactly  opposite  to  that  for  squares.  However,  it  must  be 
observed  that  any  string  of  figures  has  two  roots,  the  proper 
one  for  any  number  depending  on  its  digits.  The  square  root 
of  a  number  having  odd  digits,  as  144, 16,000  and  .000,25,  is 
found  on  the  lower  scales  under  A'  and  B',  while  for  numbers 
of  even  digits,  as  14.4,  1600  and  .0025,  the  root  is  under  A" 
and  B". 

FOR    A    NUMBER    WITH    ODD    DIGITS  FOR  A  NUMBER  WITH  EVEN  DIGITS 


Under  number 


find  square  root 


B'  A"  II     Under  number         II   B" 


C  D    ||     find  square  root 


The  square  roots  of  numbers  may  also  be  determined  with 
either  set  of  scales  by  setting  the  runner  to  the  number  and 
finding  the  number  at  the  index  of  the  slide  which  equals  that 
on  the  slide  under  the  runner. 

If  the  square  root  is  found  under  A'  or  B'  the  number  of  its 
digits  equals  (digits  in  given  number  4- 1)  Xi;  and  for  a  square 
root  under  A"  or  B",  its  digits  is  simply  one-half  those  of  the 
given  number. 


Powers  and  Roots 


25 


53.  CUBES. — The  third  power  or  cube  of  a  number  is  best 
found  by  either  of  the  following  methods: 


A 

find  cube 

B 

At  number 

c 

Set   1 

D 

to  number 

A 

to  number 

find  cube 

B 

Set  1 

C 

Over  number 

Rules  may  be  stated  for  locating  the  decimal  point  in  any 
cube,  but  it  is  better  to  determine  its  position  by  inspection 
of  the  given  number. 

54.  CUBE  ROOTS.— Cube  roots  are  best  found  with  the 
slide  inverted,  as  will  be  shown.  However,  such  may  be  de- 
termined by  setting  the  runner  to  the  given  number  on  A  and 
noting  the  number  on  D  under  the  index  on  C  equal  to  the 
number  on  B  under  the  runner.  In  this  way  three  cube  roots 
of  any.  string  of  figures  may  be  found,  the  correct  one  in  any 
case  depending  on  the  digits  of  the  given  number.  For  num- 
bers containing  — 8,  — 5,  — 2,  1,  4,  7,  etc.,  digits,  scales  A" 
and  B"  and  the  left  index  of  C  are  used;  for  numbers  of  —  7, 
—  4,  —  1,  2,  5,  8,  etc.,  digits,  scales  A'  and  B"  and  the  right 
index  of  C  are  used;  and  for  numbers  having  —  6,  —  3,  0,  3,  6, 
9,  etc.,  digits,  the  A"  and  B"  scales  are  used  with  the  right 
index  of  C.  There  is  one  digit  in  the  cube  root  for  each  period 
of  three  figures,  or  less  in  the  extreme  period,  contained  in  the 
given  number,  counting  from  the  decimal  point  toward  the  left 
for  numbers  greater  than  1,  and  toward  the  right  for  numbers 
wholly  decimal.     As  examples  of  these  principles  the  cube 

root  of  2,700  is  13.92;  \f  27,000  =  30;  and  ty  .000,27=  .0647 
It  will  be  observed  from  the  last  case  that  the  periods  in  deci- 
mal numbers  indicate  minus  digits. 

55.  HIGHER  POWERS.— The  fourth  power  of  a  number  is 
equal  to  the  square  of  its  square,  and  the  sixth  power  is  the 
square  of  its  cube  or  the  cube  of  its  square.  Other  powers 
may  be  found  in  this  way,  but  for  those  greater  than  the  fourth 
it  is  better  to  use  the  scale  of  logarithms,  as  will  be  explained 
later.  The  setting  for  the  fourth  power  is  as  follows,  the  deci- 
mal point  being  located  by  remembering  the  rules  for  squares : 


A 

find  fourth  power 

C 

Set  1 

Over  number 

D 

to  number 

56.  FRACTIONAL  POWERS.— In  almost  all  cases  such 
powers  should  be  solved  by  means  of  the  scale  of  logarithms, 
although  there  are  a  few  exceptions.  The  one-fourth  power 
of  a  number, which  is  the  same  as  its  fourth  root,  is  best  deter- 
mined by  extracting  the  square  root  of  its  square  root,  due 
attention  being  paid  to  the  digits  in  each  setting. 


26  Mannheim  Slide  Rule 

The  two-thirds  power  is  determined  by  setting  the  runner 
to  the  number  on  D,  and  then  finding  the  cube  root  of  the 
square  on  A  by  the  method  of  paragraph  54,  or  with  the 
slide  inverted,  as  will  be  shown. 

The  three-halves  power  is  obtained  as  follows,  care  being 
taken  that  the  answer  is  found  under  the  proper  scale  of  B : 


B 

Under  number 

C 

Set    1 

D 

to  number 

read  three-halves  power 

57.  POWERS  WITH  PROPORTIONAL  DIVIDERS.— Where 

a  series  of  numbers  is  to  be  raised  to  the  same  power,  whether 
whole  or  decimal,  this  method  may  be  found  useful.  Set  the 
pair  of  proportional  dividers  so  that  the  ratio  of  its  openings 
in  linear  measure  is  equal  to  the  exponent  of  the  required 
power.  Then  by  opening  one  side  of  the  dividers  from  the 
left  index  to  any  number  on  the  logarithmic  scale,  the  opening 
of  the  other  side  measured  along  the  same  scale  will  give  the 
required  power. 

IX.     INVERTED  SLIDE. 

58.  RECIPROCALS.— The  slide  may  be  turned  around  end 
for  end  so  that  scale  C  is  in  contact  with  A  and  B  with  D.  It 
will  be  observed  that  the  scales  of  the  slide  now  progress  from 
right  to  left.  With  this  arrangement  and  having  the  indices  in 
line,  all  numbers  on  C  inverted  (CI)  are  reciprocals  of  those 
directly  below  on  D,  and  similarly  for  A  and  B  inverted  (BI) . 

59.  MULTIPLICATION  AND  DIVISION.— By  setting  any 
logarithm  on  CI  over  any  logarithm  on  D,  their  sum  is  found 
on  D  under  the  index  of  CI,  and  similarly  for  A  and  BI.  Also 
the  difference  of  logarithms  is  obtained  by  placing  the  index 
of  CI  over  one  logarithm  on  D  and  reading  on  D  under  the 
other  logarithm  on  CI.     Hence  the  following: 


MULTIPLICATION. 


CI  II  Set  one  number 


opposite  other  number 


Opposite  1  IIBI 


read  product 


DIVISION. 

CI  ||  Set   1  Opposite  divisor    IIBI 

D    II  opposite  dividend  read  quotient         ||A 

Where  a  constant  quantity  is  to  be  successively  divided  by 
a  series  of  numbers,  this  method  of  division  is  valuable,  since 
all  quotients  may  then  be  read  off  without  resetting  the  slide. 

60.  INVERSE  PROPORTION.— In  such  problems,  sets  of 
factors  are  obtained  whose  products  are  equal  to  each  other. 
This  operation  is  best  accomplished  with  the  slide  inverted. 


Combined  Settings  27 

For  example,  how  many  teeth  must  a  gear  wheel  have  if  it  is 
to  turn  three  hundred  times  per  minute  when  engaging  with 
another  gear  of  48  teeth  making  50  R.  P.  M. 

CI  II  Set  50  R.  P.  M.       I   Opposite  300  R.  P.  M.   IjBI 


D    II  opposite  48  teeth      I   read  8  teeth  ||A 

61.  CUBE  ROOTS.— The  method  with  the  slide  inverted 
differs  from  that  of  paragraph  54  in  that  the  index  of  the  slide 
is  set  to  the  given  number  on  A  and  the  two  equal  numbers 
are  found  in  contact  on  BI  and  D.  The  scales  and  indices  to 
be  used  are  the  same  for  each  of  the  three  cases,  it  being  kept 
in  mind,  however,  that  B"  now  occupies  the  left-hand  side  of 
(BI) ,  and  that  the  right  and  left  indices  are  to  be  considered 
as  interchanged.  The  method  of  locating  the  decimal  point 
is  the  same  as  given  in  paragraph  54. 

X.     COMBINED  SETTINGS. 

62.  LIST  OF  SETTINGS.— The  adaptability  of  the  slide  rule 
to  the  ordinary  calculations  of  various  forms  is  exemplified 
by  the  following  list  of  useful  settings,  which  combine  the  use 
of  the  upper  and  lower  scales.  Only  those  operations  requir- 
ing a  single  setting,  but  not  especially  mentioned  before,  are 
given,  although  the  list  might  be  increased  considerably  by 
using  the  runner  for  intermediate  results  in  the  more  compli- 
cated forms  of  problems.  The  reader  will  observe  that  alter- 
native methods  of  procedure  are  possible  for  almost  every  set- 
ting, and  he  is  urged  to  become  familiar  with  them. 

In  the  following  settings  a,  b  and  c  stand  for  any  numbers 
whatsoever,  while  x  represents  the  required  quantity.  Care- 
ful attention  must  be  paid  to  the  digits  in  the  numbers  of 
which  roots  are  found.  The  decimal  point  may  be  located  in 
each  problem  by  properly  combining  the  preceding  rules,  or 
by  inspection  in  most  cases. 

SETTINGS  FOR  ONE  AND  TWO  NUMBERS. 

1 .  x  =  1  -r-  a2 — Set  1  on  C  to  a  on  D;  at  1  on  A  read  x  on  B. 

2.  x  =  1  ■*■  Va — Set  1  on  B  to  a  on  A;  at  1  on  D  read  x  on  C. 

3.  x  =a2  Xb — Set  1  on  C  to  a  on  D;  at  b  on  B  read  x  on  A. 

4.  x  =a2  + b — Set  b  on  B  over  a  on  D;  at  1  on  B  read  x  on  A. 

5.  x  =  a  H-b2 — Set  b  on  C  under  a  on  A;  at  1  on  B  read  x  on  A. 

6.  x  =a2  Xb2 — Set  1  on  C  to  a  on  D;  over  b  on  C  read  x  on  A. 

7.  x  =  a2  -s- b2 — Set  b  on  C  to  a  on  D;  at  1  on  B  read  x  on  A. 

8.  x  =a3  -nb — Set  b  on  B  to  a  on  A;  over  a  on  C  read  x  on  A. 

9.  x  =a3  ^b2 — Set  b  on  C  to  a  on  D;  at  a  on  B  read  x  on  A. 

10.  x  =a4  ^b2 — Set  b  on  C  to  a  on  D;  over  a  on  C  read  x  on  A. 

11.  x=VaXb — Set  1  on  B  to  a  on  A;  under  b  on  B  read  x 

onD. 

12.  x  =  Va  -5-b — Set  b  on  B  to  a  on  A;  at  1  on  C  read  x  on  D. 


28 


Mannheim  Slide  Rule 


13.  x=axVb — Set  1  on  Btobon  A;  at  a  on  C  read  x  on  D. 

14.  x  =  a  -t- Vt> — Set  b  on  B  over  a  on  D;  at  1  on  C  read  x  on  D. 

15.  x  =  Vaj-b— Set  b  on  C  under  a  on  A;  at  1  on  C  read  x  on  D. 

16.  x  =Va3  -nb — Set  b  on  B  to  a  on  A;  at  a  on  C  read  x  on  D. 

17.  x  =  W  -=-b — Set  b  on  C  to  a  on  D;  under  a  on  B  read  x 

onD. 

SETTINGS  FOR  THREE  NUMBERS. 

18.  x  =  a2  Xb  -s-  c — Set  c  on  B  to  b  on  A;  over  a  on  C  read  x  on  A. 

19.  x  =a2  Xb  -f-c2 — Set  c  onCto  a  onD;  at  b  on  B  read  x  on  A. 

20.  x  =a2  Xb2  vc — Set  c  on  B  over  a  on  D;  over  b  on  C  read  x 

on  A. 

21.  x=a2Xb2-rC2 — Set  conCto  a  onD;  over  b  on  C  read  x 

on  A. 

22.  x  =  VaXb  4-c — Set  c  on  B  to  a  on  A;  under  b  on  B  read 

x  on  D. 

23.  x  =  axVb-^c — Set  c  on  B  to  b  on  A;  at  a  on  C  read  x 

on  D. 

24.  x=VaXbH-c — Set  c  on  C  under  a  on  A;  under  b  on  B 

read  x  on  D. 

25.  x  =aXb  -v-Vc — Set  c  on  B  over  a  on  D;  at  b  on  C  read  x 

onD. 

26.  x  =VaXb  ^c — Set  c  on  C  to  b  on  D;  under  a  on  B  read 

x  on  D. 

XI.     SCALE  OF  SINES. 

63.  NOTATION  OF  SCALE.— On  the  back  of  the  slide  are 
three  scales  each  extending  the  full  graduated  length.  Two 
of  these  are  trigonometrical  scales  and  the  other  is  the  scale  of 
logarithms.  The  scale  of  sines  is  arranged  above  the  other 
two  and  is  marked  for  identification  by  the  letter  S.  This 
scale  is  graduated  from  approximately  0°  35'  to  90°,  as  shown  in 
Table  IV. 

TABLE  IV. 


35' 

TO 

10* 

10° 

TO 

20° 

20° 

TO 

40° 

40° 

TO 

70° 

70° 

TO 

80° 

113 
5' 

60 
10' 

20 

30' 

30 

1° 

5 

2° 

80° 

TO 

90° 


Number  of  divisions. .  . 
Value  of  each  division . 


64.  NATURAL  SINES.— If  the  slide  be  turned  over  so  that 
the  sine  scale  is  adjacent  to  scale  A  of  the  rule,  then  with  cor- 
responding indices  in  line  any  angle  on  S  is  in  contact  with  its 
natural  sine  on  A.  The  maximum  value  of  the  sine  is  1,  the 
sines  read  on  A'  having  — 1  digit,  while  those  on  A."  have  zero 
digits.     Thus  sine  3°  35'  is  .0625,  and  sine  20°  30'  is  .350. 


Scale  of  Sines  29 

The  sines  of  all  angles  may  also  be  found  with  the  slide  in 
its  normal  position  by  setting  the  angle  to  the  upper  index 
mark  in  the  right-hand  recess  in  the  base  of  the  rule.  The 
sine  is  then  read  under  either  index  of  A",  and  the  decimal 
point  is  located  by  remembering  that  angles  between  35'  and 
5°  44'  have  — 1  digit,  and  those  between  5°  45'  and  90°  have 
zero  digits.  Sines  of  angles  greater  than  90°  may  be  deter- 
mined from  the  following : 

From    90°  to  180°,  sine  a°  =sine  (180°— a°). 
From  180°  to  270°,  sine  a°=  —  sine  (a°— 180°). 
From  270°  to  360°,  sine  a°=  —  sine  (360°— a°). 

65.  SINES  OF  ANGLES  400  TO  900.— It  will  be  seen  that 
the  divisions  rapidly  diminish  in  length,  making  it  impossible 
to  obtain  accurate  readings  toward  the  end  of  the  scale. 
Hence  in  most  cases  it  is  desirable  to  calculate  the  sines  of 
angles  between  40°  and  90°  using  the  relation 

<    90°— a° 
Sine  a°  =1—2  X  sine2 ^ 

90° a° 

The  angle  k *s  se^  t°  ^ne  index  of  the  recess  and  on 

B  under  the  index  of  A"  is  read  the  corresponding  sine,  or 
x,  which  is  squared  without  shifting  the  slide  by  reading 
on  B  under  x  on  A.  This  final  value  on  B  is  then  doubled 
and  subtracted  from  1. 

90° 78°  14' 

As  an  example  determine  sine  78°  14'.     Set 2 or 

5°  53'  to  the  upper  index  of  the  right-hand  recess  and  under 
either  index  of  A"  read  .1025  on  B,  which  squared  gives  .0105 
onB'  under  .1025  on  A".  Hence  sine  78°  14'  =  1  —  2  X  .0105  or 
.979. 

66.  SINES  OF  SMALL  ANGLES.— The  sines  of  angles  less 
than  35'  are  almost  exactly  proportional  to  the  corresponding 
angles,  so  that  all  such  sines  may  be  read  directly.  For  this 
purpose  there  is  a  graduation  on  the  sine  scale  designated  (") 
and  another  one  marked  (').  For  sines  of  small  angles  ex- 
pressed in  seconds,  the  graduation  (")  is  placed  in  contact 
with  the  number  on  A  which  is  numerically  equal  to  the  given 
angle,  and  then  over  the  index  of  S  is  read  its  sine.  For  small 
angles  expressed  in  minutes  the  graduation  (')  is  used  in  the 
same  way.  In  locating  the  decimal  points  of  these  small  an- 
gles, it  must  be  noted  that  the  sine  of  angles  less  than  2"  has 
—  5  digits;  sine  3"  to  sine  20"  has  —  4  digits;  sine  21"  to  sine 
3'  26"  has  —  3  digits ;  and  sine  3'  27"  to  sine  34'  23"  has  —  2  digits. 

67.  MULTIPLICATION  AND  DIVISION  OF  SINES.— The 


30  Mannheim  Slide  Rule 

sines  of  angles  may  be  multiplied  or  divided  by  the  ordinary 
method  for  numbers,  having  scale  S  in  contact  with  A. 

TO    MULTIPLY    BY    SINE.  TO    DIVIDE    BY    SINE. 

A  1  [to  number    Iread  product  A  1 1 to  number     (read  quotient 


S  llSetindex       I  At  angle  S  llSetangle       I  At  index 

To  divide  the  sine  of  an  angle  by  any  number,  proceed  as  if 
dividing  the  number  by  the  sine,  but  then  read  the  number  on 
B  at  the  index  of  the  recess,  which  will  be  the  desired  quotient. 
The  decimal  points  in  these  problems  are  best  located  by  in- 
spection. 

68.  NATURAL  COSINES.— The  cosine  of  any  angle  a0  is 
equal  to  sine  (90°  —  a°).  Hence  by  setting  on  the  sine  scale 
for  (90° — a°),  cosine  a°  is  determined  directly;  and  it  may  be 
multiplied  or  divided  by  the  ordinary  methods  for  sines.  For 
accurately  finding  the  cosines  of  angles  less  than  50°  it  is  usu- 
ally desirable  to  employ  the  following  relation: 

a° 
Cosine  a°  =  1  —  2  X  sine2  ^' 

The  cosines  of  angles  between  89°  25'  and  90°  may  be  found 
by  means  of  the  special  graduations  (")  and  (')  on  scale  S,  as 
shown  in  paragraph  66  for  the  determination  of  sines  of  very 
small  angles. 

Cosines  of  angles  less  than  84°  15'  have  zero  digits,  while  for 
angles  between  84°  16'  and  89°  25'  there  is  one  zero  directly 
following  the  decimal  point.  The  cosines  of  angles  greater 
than  90°  may  be  determined  by  means  of  the  following: 

From    90°  to  180°,  cosine  a°  =  —  sine  (a°—  90°) . 
From  180°  to  270°,  cosine  a°  =—  sine  (270°—  a°). 
From  270°  to  360°,  cosine  a°  -  sine  (a°—  270OS> 

69.  NATURAL  SECANTS.— The  secant  of  any  angle  is  equal 
to  the  reciprocal  of  the  cosine.  Hence  for  any  angle  a°,  (90° 
—  a°)  on  S  is  set  to  the  index  of  the  recess  and  over  1  on  B  is 
read  secant  a°  on  A;  or  with  (90° —  a°)  on  S  set  to  1  on  A,  se- 
cant a°  is  read  on  A  over  the  index  of  S.  Rules  may  be  stated 
for  the  position  of  the  decimal  point,  but  it  is  advisable  to 
locate  it  by  inspection. 

70.  NATURAL  COSECANTS.— The  cosecant  of  an  angle  is 
equal  to  the  reciprocal  of  the  sine  of  that  angle,  which  may 
be  found  directly  over  the  index  of  B  with  the  angle  set  to  the 
index  of  the  recess.  If  scale  S  is  placed  in  contact  with  A,  then 
the  angle  on  S  is  set  to  the  index  of  A  and  at  the  index  of  S 
is  read  the  cosecant  on  A.  The  decimal  point  is  best  located 
by  inspection,  remembering  the  rules  for  decimals  in  sines. 

71.  NATURAL  VERSED  SINES  AND  COVERSED  SINES. 
— The  versed  sine  of  a°  equals  (1 —  cosine  a°).  while  the  co- 


Scale  of  Tangents  31 

versed  sine  of  a°  is  equal  to  (1 —  sine  a°).  These  functions  are 
found  by  obtaining  the  cosine  and  sine  respectively,  and  sub- 
tracting the  results  from  1. 

XII.     SCALE  OF  TANGENTS. 

72.  NOTATION  OF  SCALE.— Along  the  lower  edge  of  the 
under  side  of  the  slide  is  a  scale  of  tangents  marked  T.  It  is 
used  in  conjunction  with  the  lower  scales,  and  with  them  gives 
directly  the  natural  tangents  of  all  angles  from  approximately 
5°  43'  to  45°.  From  5°  45'  to  20°  there  is  a  total  of  171  divi- 
sions, each  having  a  value  of  5',  and  from  20°  to  45°  there  are 
150  divisions,  each  being  equivalent  to  10' 

73.  NATURAL  TANGENTS.— By  placing  scale  T  adjacent 
to  D  of  the  rule  and  setting  corresponding  indices  in  line,  the 
natural  tangent  of  any  angle  on  T  is  found  in  contact  on  D. 
Tangents  may  also  be  read  with  the  slide  in  its  normal  posi- 
tion by  setting  the  angle  on  T  to  the  lower  index  of  the  left- 
hand  recess  of  the  rule  and  reading  on  C  at  the  index  of  D. 
Thus  the  tangent  of  13°  30'  is  .24,  the  result  always  having 
zero  digits  for  angles  between  5°  43'  and  45°.  Tangents  of  an- 
gles greater  than  90°  may  be  determined  by  aid  of  the  follow- 
ing: 

From   90°  to  180°,  tangent  a°  =  —  tangent  (180°—  a°). 
From  180°  to  270°,  tangent  a°  = tangent  (a°—  180°). 
From  270°  to  360°,  tangent  a°  =—  tangent  (360°— a°). 

The  tangents  of  angles  less  than  5°  43'  are  very  closely  equal 
to  the  corresponding  sines,  and  hence  maybe  determined  di- 
rectly from  scale  S.  For  very  small  angles  expressed  in  sec- 
onds or  minutes,  the  special  graduations  (")  or  (')  respectively, 
on  scale  S  give  accurate  results  for  tangents  also. 

74.  TANGENTS  OF  ANGLES  45°  TO  90°.— For  these  angles 
the  natural  tangents  may  be  obtained  by  means  of  the  rela- 
tion 

Tangent  a° 


tangent  (90°—  a°) 


This  operation  is  performed  in  one  setting  by  placing  (90° —  a°) 
on  T  to  1  on  D  and  at  the  index  of  T  reading  tangent  a°  on  D. 
Tangents  of  angles  between  45°  and  84°  17'  have  1  digit;  be- 
tween 84°  18'  and  89°  25' ,  2  digits;  and  at  90°  the  tangent  is 
infinite. 

75.  MULTIPLICATION  AND  DIVISION  OF  TANGENTS.— 

Problems  of  this  kind  are  solved  by  the  ordinary  methods  of 
multiplication  and  division  as  explained  for  natural  sines  in 
paragraph  67. 

76.  NATURAL  COTANGENTS.— The  cotangent  of  an  angle 
equ?ls  the  reciprocal  of  the  corresponding  tangent.     With  the 


PART   II. 


THE   MULTIPLEX   SLIDE    RULE 


Part  II— Multiplex  Slide  Rule 


»Ul^T±t>!il>!LLTIli'in'lli^mLUi.''U'.tlL>inL 


immiii#iiR  It ItlTfTlTlTltiiiihiiTiiliiiniliiirpiiif'Y UlTfTfTlTfij  1 1 irrimfi 

f        l'l    'I'i'l'JTJ'J'lT' ,'l,MliMl.jl,MIII|lli.J|NlliIIMil|lli 

Ir-Ui-iU'l-U  *  I  ^  i'liininiiiiaiiii'iiai  i,j.i  n  \  .1  ii'j.^uuii'Hj.inijiiiwuipr'i  A  fryrpp^^| 
Pd  I  *  ■  r  i  t  .r  ,r,  r.rlT.r.T  ,1  i.,  .it  i  i.i  .it  ■■ ,  ,t, ,,  .I.m.t.m  JMM?l,llTMlff,„,T,,,ffi,il,(n,,,i,,[fB,.,^4 

I.     INTRODUCTION. 

1.  APPLICATION. — As  its  name  implies,  the  Multiplex  slide 
rule  is  a  calculating  instrument  of  many  uses.  Not  only  does 
it  broaden  the  field  of  application,  but  it  also  offers  a  conven- 
ient means  of  more  rapid  working,  securing  greater  accuracy 
at  the  same  time.  The  theoretical  and  mechanical  principles 
upon  which  the  Multiplex  slide  rule  is  based,  are  identical  in 
all  respects  to  those  underlying  the  action  of  the  ordinary 
Mannheim  rules,  so  that  the  operator  has  but  little  more  to 
learn,  although  there  is  much  to  be  gained. 

Not  only  does  the  Multiplex  solve  all  the  arithmetical,  trig- 
onometrical and  logarithmic  examples  which  are  possible  with 
the  Mannheim,  and  in  the  same  convenient  and  rapid  manner, 
but  it  further  possesses  the  following  characteristic  advan- 
tages : 

1.  Multiplication  of  three  numbers  in  one  setting. 

2.  Division  of  one  number  by  two  numbers  in  one  setting. 

3.  More  convenient  solution  of  inverse  proportion. 

4.  Direct  solution  in  a  single  setting  of  a  series  of  divisions 
with  a  constant  dividend. 

5.  Direct  reading  of  cubes  and  cube  roots. 

6.  Direct  reading  of  three-halves  and  two-thirds  powers. 

7.  Direct  solution  in  a  single  setting  of  many  combined 
operations  which  require  the  slide  to  be  shifted  with  the  Mann- 
heim rule. 

2.  ACCURACY. — The  division  of  all  scales  of  our  rules  are 
practically  perfect,  so  that  the  only  errors  which  enter  into 
the  results  of  calculations  are  those  due  to  setting  and  read- 
ing. Hence  by  reducing  the  number  of  times  that  the  slide 
must  be  set  and  the  scales  read,  the  result  of  any  calculation 
becomes  more  accurate,  even  where  shorter  scales  are  em- 
ployed. 

3.  SAVING  IN  TIME.— Next  to  accuracy  of  results  the  cal- 
culator is  most  concerned  with  rapidity  in  working.  Here 
again  the  Multiplex  rule  is  superior  to  all  others,  since  it  elimi- 
nates some  of  the  settings  and  readings  required  in  many  of 
the  common  problems. 


38 


Multiplex  Slide  Rule 


-6650 


■8040 


=jjj ?43Q 


;-= &y» 


4190 


— 5SM 


- 5060 


MECHANICAL  ADVANTAGES.— Due  to  improved  de- 
sign the  Multiplex  is  the  most  perfect 
acting  and  most  durable  of  all  types  of 
slide  rules.  A  study  of  the  following 
text  will  reveal  to  the  reader  a  simple 
instrument,  of  wide  application,  cap- 
able of  rapid  and  accurate  results,  and 
one  of  superior  and  lasting  mechanical 
qualities. 

5.  NOTE. — All  processes  of  multipli- 
cation, division,  proportion,  powers, 
roots  and  logarithms,  together  with  the 
determination  of  trigonometrical  func- 
tions and  combined  operations,  may  be 
solved  with  the  Multiplex  just  as  with 
the  Mannheim.  Hence  in  the  following 
text  only  those  settings  which  are  ap- 
plicable to  the  Multiplex  alone  will  be 
considered,  the  other  features  and  prin- 
ciples common  to  both  types  of  rules 
being  understood  from  Part  I. 

II.     CONSTRUCTION. 

6.  GENERAL.— The  mechanical  con- 
struction of  the  Multiplex  is  in  general 
similar  to  that  of  the  Mannheim,  there 
being  the  same  arrangement  of  body, 
slide  and  runner.  We  now  manufac- 
ture three  sizes  of  each  of  two  forms  of 
the  Multiplex,  one  form  having  a  re- 
ciprocal scale  but  no  cube  scale,  while 
the  other  has  both  of  these  features. 
In  the  latter  type  the  cube  scale  is 
located  on  the  base  between  the  two 
fixed  rules,  thus  occupying  the  space 
beneath  the  slide,  which  in  other  types 
of  rules  is  devoted  to  no  useful  pur- 
pose. The  addition  of  the  cube  scale 
and  the  substitution  of  the  reciprocal 
scale  for  B'  are  the  only  differences 
between  the  scales  of  the  Multiplex 
and  the  Mannheim  slide  rules.  The 
scales  of  sines,  tangents  and  loga- 
rithms are  identical  in  arrangement 
and  use  for  both  types. 

7.  SLIDE. — For  that  form  of  the 
Multiplex  rule  which  includes  the  cube 
scale,  the  slide  has  a  recess  provided 
at  each  end  through  which  readings  are 


Construction 


39 


made  on  the  cube  scale.  Each  recess  is  spanned  by  a  strip  of 
metal,  thus  strengthening  the  mechanical  construction  and 
at  the  same  time  providing  a  convenient  means  for  drawing 
out  the  slide. 

8.  RECIPROCAL  SCALE.— Along  the  left-hand  side  of  the 
upper  edge  of  the  slide  there  is  a  complete  logarithmic  scale 
progressing  in  the  reversed  direction,  from  the  center  index  of 
B  toward  the  left.  This  scale  will  be  called  the  reciprocal 
scale  of  B  or  Br.  It  is  exactly  one-half  the  length  of  C  or  D, 
and  hence  equal  to  that  of  A',  A"  or  B".  Scale  Br  and  B"  are 
also  identical  in  every  other  respect  except  that  they  progress 
in  opposite  directions  from  the  center  index  of  B.  In  order 
to  avoid  any  possible  confusion,  the  numbers  on  scale  Br  are 
colored  red.  In  this  way  the  attention  of  the  operator  is  im- 
mediately drawn  to  the  fact  that  the  scale  in  question  pro- 
gresses in  the  reversed  direction,  from  right  to  left. 

9.  CUBE  SCALE. — This  scale  is  located  on  the  base  between 
the  fixed  rules  and  underneath  the  slide.  It  is  composed  of 
three  identical  and  complete  logarithmic  scales,  E',  E"  and 
E"',  progressing  from  left  to  right,  and  each  being  exactly  one- 
third  the  length  of  C  or  D  and  two-thirds  that  of  any  one  of 
the  upper  scales.  The  outer  indices  of  E  are  accurately  in  line 
with  those  of  the  fixed  rules,  while  the  indicator  lines  (I.  L.) 
of  the  cube  scale  are  carried  in  the  recesses  of  the  slide  directly 
in  line  with  the  corresponding  indices  of  scales  B  and  C. 

10.  READING  SCALE  Br. — The  division  of  the  reciprocal 
scale  is  identical  with  that  of  the  other  upper  scales,  as  shown 
in  Table  I  for  its  right-hand  index  equal  to  1. 

TABLE  I. 


FIVE  AND  TEN-INCH 
RULES- 

TWENTY-INCH  RULE 

10  TO  5 

5  to  2 

2  TO  1 

10  TO  4 

4  to  2 

2  TO  1 

Value  of   each  main  divi- 

1.00 

*.io 

50 

1.00 

.10 
.05 
60 

1.00 

.10 
.02 
50 

1.00 

.10 
.05 

120 

1.00 

.10 
.02 
100 

1.00 

Value  of  each  minor  divi- 

.10 

Value  of  smallest  division . 
Total  number  of  divisions.. 

01 
100 

A  little  practice  will  accustom  the  reader  to  accurately  and 
quickly  read  the  reversed  graduations  and  to  readily  pass  from 
this  scale  to  others  progressing  in  the  opposite  direction.  Fig. 
1  (see  page  38)  should  be  carefully  examined  and  compared 
with  the  ten-inch  rule,  and  then  the  following  settings  and 
readings  should  be  performed  with  the  reciprocal  scale.  Set  3 
on  Br  to  4  on  A'  and  at  1  on  Br  read  12  on  A;  note  that  3 
on  A'  and  4  on  Br  are  likewise  in  contact,  and  that  12  on  Br 
is  at  the  center  index  of  A.  Set  125  on  Br  to  36  on  A'  and  at 
1  on  Br  read  4500  on  A'.  Set  1  on  Br  to  884  on  A"  and  at 
425  on  Br  read  2.08  on  A", 


40  Multiplex  Slide  Rule 

11.  READING  SCALE  E.— Scale  E  consists  of  the  three 
parts  E',  E"  and  E'",  the  first  being  taken  on  the  extreme  left 
and  E"'  on  the  extreme  right.  Each  of  these  scales  progresses 
from  left  to  right  and  is  divided  in  exactly  the  same  way  as 
the  upper  scales  of  the  slide  and  rule. 

III.     MULTIPLICATION. 

12.  MECHANICAL  PRINCIPLES.— By  drawing  two  linear 
scales  with  equal  divisions  throughout,  but  progressing  in  op- 
posite directions,  the  addition  of  any  number  of  units  on  one 
with  those  on  the  other  may  be  obtained  mechanically  by  set- 
ting the  two  quantities  together,  as  shown  in  Fig.  2.     With  0 


FIG.  2.    ADDITION  AND  SUBTRACTION  WITH  REVERSED  SCALE. 

on  Br  set  to  7  on  A,  it  will  be  observed  that  any  two  numbers 
on  the  scales  whose  sum  is  7  are  in  contact. 

13.  TWO  FACTORS.— From  the  simple  principle  of  the  pre- 
ceding paragraph  it  will  be  evident  that  by  setting  the  loga- 
rithm of  any  number  on  Br  to  the  logarithm  of  any  number  on 
A,  their  sum,  which  is  the  logarithm  of  the  corresponding 
product,  is  found  on  A  at  the  index  of  Br.  In  the  manner 
shown  in  the  tabular  statement  below,  the  logarithms  are 
added  exactly  as  with  scales  progressing  in  the  same  direction. 

A    II  to  second  number     I  read  product 

Br  II  Set  one  number        I  At  1 

14.  THREE  FACTORS.— One  of  the  many  useful  advan- 
tages of  the  Multiplex  rule  is  multiplication  of  three  numbers 
with  but  one  setting  of  the  slide.  It  will  be  observed  that  the 
product  of  two  factors,  using  scale  Br,  is  found  on  A  at  the 
index  of  Br,  which  is  then  the  required  setting  for  the  multi- 
plication of  that  product  by  any  third  factor  on  B"  within  the 
range  of  contact. 

A'  I  to  second  number    II  A  I  read  final  product 
Br  I  Set  one  number        II  B"l  At  third  number 

The  first  and  second  factors  are  always  taken  on  Br  and  A', 
from  which  it  will  be  observed  that  the  logarithms  of  the  three 
numbers  are  directly  added  in  all  cases. 

15.  CONSTANT  PRODUCT.— If  the  index  of  Br  or  B"  be 
set  to  a  given  number  on  A,  all  combinations  of  two  factors 
whose  products  equal  the  given  number  will  be  found  in  con- 
tact between  Br  and  A.     Similarly  if  the  product  of  three  fac- 


MULTIPLIC  \TION 


41 


tors  is  to  be  a  constant  quantity,  then  all  sets  of  its  factors 
may  be  determined  by  setting  the  runner  to  the  given  quan- 
tity on  A"  and  bringing  any  number  on  B"  to  the  hair-line, 
when  the  other  two  correspond  ing  factors  will  be  in  contact  on 
Br  and  A.  These  principles  are  of  much  importance  in  en- 
gineering design  work  and  are  characteristic  features  of  the 
Multiplex  slide  rule. 

16.  PROPORTION.— Direct  proportion  is  best  solved  with 
the  scales  C  and  D  or  A  and  L".  Inverse  proportion  is  most 
conveniently  performed  by  means  of  the  reciprocal  scale  of  the 
Multiplex  rule,  the  method  b(  mg  similar  to  that  for  the  Mann- 
heim rule  with  its  slide  inverted.  As  an  example,  assume  that 
the  electrical  resistance  of  1000  feet  of  copper  wire  having  a 
cross-section  of  350,000  circular  mils  in  .030  ohm.  What  is 
the  resistance  of  1000  feet  of  copper  wire  of  700,000  circular 
mils  section? 


to   .030  ohms 


read   .015  ohms 


Br  II   Set  350,000  circular  mils    |  At  700,000  circular  mils 

By  this  method  the  problem  is  solved  as  if  dealing  with  direct 
proportion  on  the  other  scales. 

17.  DECIMAL  POINT.— The  rules  for  locating  the  decimal 
point  in  a  product  obtained  by  using  the  reciprocal  scale  are 
simple  and  may  be  remembered  as  follows: 

If  the  first  significant  figure  of  the  product  of  two  numbers 
is  greater  than  the  first  significant  figures  of  both  factors,  then 
the  number  of  digits  in  the  product  is  one  less  than  the  sum  of 
the  digits  in  the  two  factors;  if  less,  the  digits  in  the  product 
is  simply  equal  to  their  sum.  Where  the  first  significant  fig- 
ures are  the  same,  the  following  figures  must  be  likewise  com- 
pared. 

The  number  of  digits  in  the  product  of  three  factors  ob- 
tained in  one  setting  is  two  less  than  the  sum  of  those  in  the 
three  factors  if  the  final  product  is  read  on  A',  and  is  one  less 
than  their  sum  if  read  on  A". 

In  multiplying  more  than  three  factors  together  the  above 
rules  are  combined  for  the  digits  in  the  product,  or  the  decimal 
•point  is  located  by  inspection. 

18.  EXAMPLES.— The  reader  should  work  the  following 
examples  using  scale  Br  and  then  compare  the  operation  in 
each  case  with  the  process  required  for  the  lower  scales: 


EXAMPLE 

SUM  OF 
DIGITS 

FINAL  PROD- 
UCT 
READ     ON 
SCALE 

DIGITS  IN 
PRODUCT 

ANSWER 

24X  1.42  X  18.2 

182  X  2.95  X  .087 

.0024X  .56  X  7.1 

5 
3 

-1 

A' 
A" 
A" 

3 
2 

-2 

620 
46.7 
.00954 

42  Multiplex  Slide  Rule 


IV.     DIVISION. 

19.  MECHANICAL  PRINCIPLES.— Using  linear  scales  pro- 
gressing in  opposite  directions,  any  number  may  be  subtracted 
from  another  by  setting  one  index  to  the  latter  number  on  the 
other  scale,  and  reading  at  the  number  to  be  subtracted.  In 
Fig.  2  (see  page  40)  the  index  of  Br  is  set  to  7  on  A,  and  at  2 
on  Br  is  5  on  A;  and  similarly  for  1  and  6,  3  and  4,  etc.  In 
the  same  way  any  two  logarithms  on  the  slide  rule  may  be 
subtracted,  giving  the  logarithm  of  the  quotient  of  the  cor- 
responding numbers.     Hence  for  division, 


A 

Br 


to  dividend         I  read  quotient     II  Br 


Set   1  |  At  divisor  II  A 


20.  CONSTANT  DIVIDEND.— If  a  constant  quantity  is  to 
be  divided  by  a  series  of  numbers  the  entire  set  of  quotients 
may  be  read  off  directly  without  shifting  the  slide.  The  in- 
dex of  B  is  set  to  the  constant  dividend  on  A  and  at  the  divis- 
ors on  Br  are  read  in  turn  the  corresponding  quotients  on  A. 

21.  RECIPROCALS.— The  reciprocals  of  all  numbers  may 
be  read  directly  on  the  Multiplex  rule  without  shifting  the 
slide.  For  this  operation  the  index  on  Br  is  set  to  the  index 
on  A  and  at  any  number  on  Br  is  read  its  reciprocal  on  A,  or 
dee  versa. 

22.  CONTINUED  DIVISION.— Any  number  may  be  divided 
by  two  numbers  in  one  setting  of  the  Multiplex  rule.  One 
divisor  on  B"  is  set  to  the  dividend  on  A"  and  at  the  other 
divisor  on  Br  is  read  the  final  quotient  on  A. 


A" 


B" 


to  dividend        II  A  I  read  final  quotient 


Set  one  divisor  II  Br|  At  other  divisor 


By  this  method  the  sum  of  the  logarithms  of  the  two  divisors 
is  directly  subtracted  from  the  logarithm  of  the  dividend. 

23.  DECIMAL  POINT.— The  number  of  digits  in  a  quotient 
may  be  determined  as  follows : 

If  the  first  significant  figure  of  the  divisor  is  greater  than 
that  of  the  dividend,  the  number  of  digits  in  the  quotient  of 
the  two  numbers  is  equal  to  the  digits  in  the  dividend  less  those 
in  the  divisor;  if  less,  then  one  digit  must  be  added  to  this 
difference.  The  following  significant  figures  are  to  be  com- 
pared where  the  first  ones  are  alike. 

Where  one  number  is  divided  by  two  divisors,  the  number 
of  digits  in  the  final  quotient  is  two  more  than  the  difference 
between  the  digits  of  the  dividend  and  divisors  if  the  final 
quotient  is  read  on  A",  and  one  more  than  their  difference  if 
read  on  A'. 


Powers  and  Roots 


43 


24.  EXAMPLES. — The  reader  should  solve  the  following 
problems  using  scale  Br,  and  then  compare  this  method  with 
the  operations  required  for  the  lower  scales: 


DIVIDEND 

FIRST 
DIVISOR 

SECOND 
DIVISOR 

DIFFER- 
ENCE OF 
DIGITS 

FINAL 
QUO- 
TIENT 
READ    ON 
SCALE 

DIGITS 
IN    FINAL 
QUO- 
TIENT 

ANSWER 

985 

.0046 
6200 

.023 
.325 

8.4 

168 
.199 
3.95 

1 

-2 
2 

A" 
A' 
A' 

3 

—  1 
3 

255 
.0712 

187 

V.     POWERS  AND  ROOTS. 

25.  GENERAL.— Since  the  scales  A',  A",  B",  C  and  D  of  the 
two  types  of  rules  are  identical,  all  problems  and  settings  in- 
volving them  may  be  performed  in  exactly  the  same  way. 
Hence  almost  all  the  cases  given  in  Chapter  VIII  of  Part  I 
covering  powers  and  roots  apply  equally  well  to  the  Multiplex 
rule.  However,  the  following  solutions  are  best  determined 
with  the  Multiplex  and  will  be  found  of  much  value  to  the 
calculator. 

26.  SOLUTION  OF  -^-.-Problems  of  this  form  may  be  solved 

directly  by  setting  the  reciprocal  scale  over  a  on  D,  having 
the  proper  indices  of  Br  and  A  in  line,  and  then  reading  on  Br 

above  a  on  D.     For  numbers  less  than  3162.  . .  .—j-  may  be 

read  on  Br  directly  above  a  on  C.     The  decimal  point  in  such 
problems  is  best  located  by  inspection. 

27.  SOLUTION  OF    r  -    .—Much   saving  in  time  may  be 

Va 
effected  in  a  series  of  problems  of  this  kind  by  using  the  Mul- 
tiplex rule.     For  numbers  having  odd  digits,  the  left  index  of 
Br  is  set  to  the  center  index  of  A  and  under  a  on  Br  is  read 

r-  on  D.  For  numbers  having  even  digits,  r~-  may  be 
Va  Va 

found  on  C  directly  under  a  on  Br. 

28.  CUBES.— Since  scale  D  is  three  times  the  length  of  each 
of  the  scales  comprising  E,  the  logarithm  of  any  number  on  D 
is  directly  in  line  with  three  times  that  logarithm  on  E.  Hence 
any  number  on  D  is  in  line  with  its  cube  or  third  power  on  E. 

E  At  I.  L.  on  slide  read  cube  of  number 


Set  1 


D  |   to  number 

If  read  on  E',  the  cube  has  two  less  than  three  times  the 
digits  in  the  given  number;  if  read  on  E",  it  has  one  digit  less; 


44  Multiplex  Slide  Rule 

and  where  read  on  E'",  the  digits  in  the  cube  simply  equal 
three  times  the  digits  of  the  given  number.  Hence  the  cube 
of  .16  is  .0041;  423  =74,100;  and  763  =439,000. 

29.  CUBE  ROOTS.— The  cube  root  of  a  number,  which  is 
the  same  as  its  one-third  power,  may  be  read  directly  by  refer- 
ring from  E  to  D  as  follows: 


E 

Set  I.  L.  on  slide  to  number 

C 

At  1 

D 

read  cube  root  of  number 

There  are  three  cube  roots  of  any  string  of  figures,  the  proper 
one  for  any  given  number  depending  on  its  digits.  For  num- 
bers containing  —  8,  —  5,  —  2, 1,  4,  7,  etc.,  digits,  scale  E/  is 
used;  for  numbers  of  —  7,  — 4,  —  1,  2,  5,  8,  etc.,  digits,  scale 
E"  is  used;  and  for  numbers  having  —  6,  —  3,  0,  3,  6,  9,  etc., 
digits,  the  given  number  is  taken  on  E'".  There  is  one  digit 
in  the  cube  root  for  each  period  of  three  figures,  or  less  in  the 
extreme  period,  contained  in  the  given  number,  counting 
from  the  decimal  point  toward  the  left  for  numbers  greater 
than  1,  and  toward  the  right  for  decimals.  The  periods  in  deci- 
mals indicate  minus  digits.  With  these  principles  in  mind  the 

cube  root  of  9  is  seen  to  be  2.08;  V43,000  =  35 ;  V.000,125  =  .05. 

30.  THREE-HALVES  POWERS.— Since  the  length  of  scale 
A'  or  A"  is  exactly  three-halves  times  that  of  either  E',  E"  or 
E'",  the  three-halves  power  of  any  number  may  be  determined 
directly  with  the  Multiplex  rule  by  passing  from  A  to  E. 
to  number 


Set  1 


At  I.  L.  on  slide  read  three-halves  power  of  number 
In  this  process  numbers  having  odd  digits  are  taken  on  A', 
while  those  with  even  digits  are  taken  on  A". 

The  decimal  point  in  the  three-halves  power  may  be  located 
by  the  following.  Representing  the  number  of  digits  in  the 
given  number  by  Nn  and  those  in  the  power  by  Np, 

For  power  on  E',  Np  =  I  Nn  X  -~  I  — ?  •=> 

For  power  on  E"  and  number  on  A',  Np  =  I  Nn  X-~- 1  +•«"• 

For  power  on  E",  and  number  on  A",  Np  =  I  Nn  X-~  I  —  1 
For  power  on  E'",  Np  =Nn  X-^-. 

As  examples,  the  three-halves  power  of  4  is  8;  (933)3/2  = 
28,500;  V.00153=  .000,058;  and   V663-  536. 


Settings  for  the  Multiplex  Slide  Rule  45 

31.  TWO-THIRDS  POWERS.— This  operation  is  the  reverse 
of  the  preceding,  as  shown  below: 

j   read  two-thirds  power  of  number 


At  1 


A 
B 
E   | Set  l.L.  on  slide  to  number 

Numbers  having  —  8,  —  5,  —  2,  1,  4,  7,  etc.,  digits,  are  taken 
on  E';  for  —  7,  —  4,  —  1,  2,  5,  8,  etc.  digits,  use  scale  E";  and 
for  —  6,  —  3,  0,  3,  6, 9,  etc.,  digits,  the  number  is  taken  on  E'". 

By  referring  to  paragraph  29  it  will  be  seen  that  the  proper 
scale  of  E  to  use  in  finding  the  two-thirds  power  of  a  given 
number  is  the  same  as  for  its  cube  root.  Hence  both  the  one- 
third  and  two-thirds  powers  of  any  number  may  be  determined 
from  the  same  setting;  and  similarly  for  the  second  and  third 
powers,  and  for  the  one-half  and  three-halves  powers. 

The  number  of  digits  in  the  two-thirds  power  may  be  found 
as  follows,  wherein  the  same  notation  is  used  as  in  the  pre- 
ceding paragraph: 

For  number  on  E',  Np  =  I  Nn  X-~-  J  +  -~. 

For  number  on  E"  and  power  on  A',  Np  =  I  Nn  X   -  J  - 


For  number  on  E"  and  power  on  A",  Np  =(NnXv)  + 
For  number  on  E'",  Np  =  Nn  X  -^-. 


32.  OTHER  POWERS  AND  ROOTS.— By  properly  combin- 
ing the  scales  of  the  Multiplex  and  using  the  runner  for  inter- 
mediate results,  a  great  variety  of  powers  and  roots  may  be 
readily  determined.  For  a  series  of  examples  involving  the 
same  process  such  methods  are  recommended,  but  for  a  single 
problem  it  is  sometimes  better  to  resort  to  the  scale  of  loga- 
rithms on  the  back  of  the  slide,  using  the  method  outlined  in 
Chapter  XIII  of  Part  I. 

VI.     SETTINGS    FOR   THE    MULTIPLEX   SLIDE 
RULE. 

33.  LIST  OF  SETTINGS.— The  following  settings  are  in- 
tended to  supplement  those  of  Chapter  X  Part  I,  all  of  the 
latter  being  applicable  to  the  Multiplex  rule  also.  The  addi- 
tional value  of  the  Multiplex  slide  rule  may  be  judged  from 
the  great  variety  of  important  operations  which  may  be  solved 
in  a  single  setting.  By  using  the  runner  for  intermediate  re- 
sults and  settings,  the  list  may  be  enlarged  to  an  almost  un- 
limited extent. 


46  Multiplex  Slide  Rule 

The  operator  must  pay  attention  to  the  digits  in  the  given 
number  and  the  intermediate  results,  in  order  to  determine 
which  part  of  the  upper  and  cube  scales  to  use  in  any  case, 
The  decimal  point  in  the  final  result  is  usually  best  determined 
by  inspection,  although  the  preceding  rules  for  digits  may  be 
combined  for  the  purpose. 

The  letters  a,  b  and  c  represent  any  numbers  whatsoever, 
while  x  is  used  for  the  required  result.  It  will  be  remembered 
that  I.  L.  stands  for  the  indicator  line  of  the  cube  scale,  while 
CI,  B'T  and  BrI  designate  scales  C,  B"  and  Br  respectively, 
with  the  slide  inverted. 

SETTINGS  FOR  ONE  NUMBER. 

1.  x  =a2 — Set  1  on  C  to  a  on  D;  at  1  on  B  read  x  on  A. 

2.  x  =a3 — Set  1  on  C  to  a  on  D;  at  I.  L.  read  x  on  E. 

3.  x  =a4 — Set  1  on  C  to  a  on  D;  over  a  on  C  read  x  on  A. 

4.  x  =  a5 — Set  a  on  CI  over  a  on  D;  over  a  on  BrI  read  x  on  A. 

5.  x=a6 — Set  a  on  CI  over  a  on  D;  at  I.  L.  read  x  on  E. 

6.  x  =  1  -*-  a2 — Set  1  on  B  to  1  on  A;  over  a  on  D  read  x  on  Br. 

7.  x  =  1  +  a3 — Set  a  on  Br  over  a  on  D;  at  1  on  A  read  x  on  B". 

8.  x  =  1  -;-  a4 — Set  a  on  CI  over  a  on D;  at  1  on D  read  x  on  BrI. 

9.  x  =  V  a — Set  1  on  B  to  a  on  A;  at  1  on  C  read  x  on  D. 

10.  x  =  Va3 — Set  1  on  B  to  a  on  A;  at  I.  L.  read  x  on  E. 

11.  x  =  Va* — Set  a  on  CI  over  a  on  D;  at  a  on  BrI  read  x  on  D. 

12.  x=  va9 — Set  a  on  Br  over  a  on  D;  at  I.  L.  read  x  on  E. 

13.  x  =  Va— Set  I.  L.  to  a  on  E;  at  1  on  C  read  x  on  D. 

14.  x  =  Va2 — Set  I.  L.  to  a  on  E;  at  1  on  B  read  x  on  A. 

15.  x  =  Va4 — Set  I.  L.  to  a  on  E;  at  a  on  Cread  x  on  D. 

16.  x  =  V3^ — Set  I.  L.  to  a  on  E;  at  a  on  B"  read  x  on  A. 

17.  x  =  Va8 — Set  I.  L.  to  a  on  E;  over  a  onC  read  x  on  A. 

18.  x  =  Va5 — Set  I.  L.  to  a  on  E;  under  a  on  B"read  x  on  D. 

19.  x  =  1  -T-  Va — Set  I.  L.  to  a  on  E;  at  1  on  D  read  x  on  C. 

20.  x  =  1  4-  Va2 — Set  I.  L.  to  a  on  E;  at  1  on  A  read  x  on  B". 

21.  x  =  1  -r-  Va4 — Set  I.  L.  to  a  on  E;  at  a  on  CI  read  x  on  A. 

22.  x  =  1  ■*■  Va — Set  I.  L.  to  a  on  E;  under  a  on  Br  read  x  on  D. 

SETTINGS  FOR  TWO  NUMBERS. 

23.  x  =a  Xb — Set  a  on  Br  to  b  on  A;  at  1  on  B  read  x  on  A. 

24.  x  =  a  -f-b — Set  1  on  B  to  a  on  A;  at  b  on  Br  read  x  on  A. 

25.  x  =a  Xb2 — Set  1  on  B  to  a  on  A;  over  b  on  C  read  x  on  A. 

26.  x  =  a  ^b2 — Set  b  on  C  under  a  on  A;  at  1  on  B  read  x  on  A. 

27.  x  =a2  -J-b — Set  1  on  C  to  a  on  D;  at  b  on  Br  read  x  on  A. 

28.  x  =a2  Xb2 — Set  1  on  C  to  a  on  D;  over  b  on  C  read  x  on  A. 

29.  x  =a2  -^b2 — Set  b  on  C  to  a  on  D;  at  1  on  B  read  x  on  A. 

30.  x  =a3  Xb — SetaonB'T  to  a  onD;  over  b  on  BrI  readxonA. 
gl .  x  =  a3  ■*-  b — Set  a  on  B"I  to  a  on  D ;  over  b  on  B"I  read  x  on  A. 
32.  x=a-fb3 — Set  b  on  C  under  a  on  A";  at  b  on  Br  read  xonA. 


34. 

x 

35. 

X 

36. 

X 

37. 

X 

38. 

X 

39. 

X 

40. 

X 

41. 

X 

42. 

X 

Settings  for  the  Multiplex  Slide  Rule  47 

33.  x  =a3  Xb2 — Set  a  on  CI  over  b  on  D;  over  a  on  BrI  readx 

on  A. 
=  a3  -^b2 — Set  b  on  C  to  a  on  D;  at  a  on  B"  read  x  on  A. 
=  a2  -^b3 — Set  b  on  C  to  a  on  D;  at  b  on  Br  read  x  on  A. 
=  a3  Xb3 — Set  a  on  CI  over  b  on  D;  at  I.  L.  readx  on  E. 
=  a3-7-b3 — Set  b  on  C  to  a  on  D;  at  I.  L.  read  x  on  E. 
=  a4  Xb — Set  a  on  CI  over  a  on  D;  over  b  on  BrI  read  x 

on  A. 
=  a4  +  b — Set  a  on  CI  over  a  on  D ;  over  b  on  B"I  read  x  on  A. 
=  a  H-b4 — Set  b  on  CI  over  b  on  D;  under  a  on  A  read  x 

on  BrI. 
=  a4  -^b2 — Set  b  on  C  to  a  on  D;  over  a  on  C  read  x  on  A. 
=  1  -s-  (a  Xb) — Set  a  on  Br  to  b  on  A;  at  1  on  A  read  x 

on  B". 

43.  1  +  (a2  Xb) — Set  b  on  Br  over  a  on  D;  at  1  on  A  read  x 

on  B". 

44.  x  =  \+  (a2  Xb2) — Set  a  on  CI  over  b  on  D;  at  1  on  Dread 
x  on  BrI. 

=  Va  Xb — Set  a  on  Br  to  b  on  A;  at  1  on  C  read  x  on  D. 
=  Va  -s-  b — Set  b  on  B"  to  a  on  A;  at  1  on  C  read  x  on  D. 
=  axVb— SetlonC  to  a  on  D ;  under  b  on  B"  read  xonD. 
=  a  -r-  Vb — Set  b  on  B"  over  a  on  D;  at  1  on  C  read  x  on  D. 
=  Va  -r  b — Set  b  on  C  under  a  on  A;  at  1  on  C  read  x  onD. 
=  a  X  Vb3 — Set  b  on  Br  over  a  on  E;  at  I.  L.  read  x  on  E. 
=  a  +  Vb3 — Set  b  on  B"  over  a  on  E;  at  I.L.  read  x  on  E. 
=  a3  X  Vb3 — Set  b  on  Br  over  a  on  D;  at  I.  L.  read  x  on  E. 
=  a3  -j-  Vb3 — Set  b  on  B"  over  a  on  D ;  at  I.  L.  read  x  on  E. 
-  Va3  j-  b3 — Set  b  on  C  under  a  on  A;  at  I.  L.  read  x  on  E. 
=  Va3  Xb3 — Set  a  on  Br  to  b  on  A;  at  I.  L.  read  x  on  E. 
=  Va3  -^b3 — Set  b  on  B"  to  a  on  A;  at  I.  L.  read  x  on  E. 
=  a  X  Vb — Set  I.  L.  to  b  on  E;  at  a  on  C  read  x  on  D. 
=  a  +  Vb — Set  a  on  C  over  b  on  E;  at  1  on  D  read  x  on  C. 
=  Va  -v-b — Set  I.  L.  to  a  on  E;  under  b  on  CI  read  x  on  D- 
=  a  X  Vb2 — Set  I.  L.  to  b  on  E;  at  a  on  B"  read  x  on  A. 
=  Va2  +h — Set  I.  L.  to  a  on  E;  at  b  on  Br  read  x  on  A. 
-an-Vb2 — Seta  on  B"  over  b  on  E;  at  1  on  A  read  x 

on  B". 
=  a2  X  Vb2 — Set  I.  L.  to  b  on  E;  over  a  on  Cread  x  on  A. 
=  Va2  -^b2 — Set  I.  L.  to  a  on  E;  at  b  on  CI  read  x  on  A. 
=  a2  -f-  Vb2 — Set  a  on  C  over  b  on  E;  at  1  on  A  read  x 

on  B". 
=  1  h- Va  Xb — Set  a  on  Br  to  b  on  A;  at  1  on  D  read  x 

onC. 
=  1  -r-  (a  X  Vb) — Set  a  on  C  to  1  on  D;  under  b  on  Br  read 

x  on  D. 


45. 

X 

46. 

X 

47. 

X 

48. 

X 

49. 

X 

50. 

x 

51. 

X 

52. 

X 

53. 

X 

54. 

X 

55. 

X 

56. 

X 

57. 

X 

58. 

X 

59. 

X 

60. 

X 

61. 

X 

62. 

X 

63. 

X 

64. 

X 

65. 

X 

66. 

X 

67. 

X 

48  Multiplex  Slide  Rule 

SETTINGS  FOR  THREE  NUMBERS. 

68.  x=aXbXc — Set  a  on  Br  to  b  on  A';  at  con  B"  read  x  on  A. 

69.  x=aXb-^-c — Set  a  onBrtobonA;  at  c on  Br  read  x  on  A. 
70i.  x  =a  -^  (b  Xc) — Set  b  on  B"  to  a  on  A";  at  c  on  Br  read  x 

on  A. 

71.  x  =  a  Xb  Xc2 — Set  a  on  Br  to  b  on  A;  over  c  onC  readxon  A. 

72.  x  =a2  Xb .+  c — Set  b  on  Br  over  a  on  D;  at  c  on  Br  read  x 

on  A. 

73.  x=aXbv  c2 — Set  c  on  C  under  a  on  A;  at  b  on  B"  read  x 

on  A. 

74.  x  =a2  -s-  (b  Xc) — Set  b  on  B"  over  a  on  D;  at  c  on  Br  read 

x  on  A. 

75.  x  =a2  Xb2  -r-c — Set  c  on  B"  over  a  on  D;  over  b  on  C  read 

x  on  A. 

76.  x  =a2  Xb  4-  c2 — Set  c  on  C  to  a  onD;  at  b  onB"  read  x  on  A. 

77.  x  =  a2Xb2-^-c2 — Set  c  on  C  to  a  on  D;  over  b  on  C  read  x  onA. 

78.  x  =  a3  Xb3  -f-  c3 — Set  c  on  C  to  a  on  D;  under  b  on  C  read  x 

onE. 

79.  x  =  Va  Xb  Xc — Set  a  on  Br  to  b  on  A';  under  c  on  B"  read 

x  on  D. 

80.  x  =  Va  Xb  h-  c — Set  a  on  Br  to  b  on  A;  under  c  on  Br  read 

x  on  D. 

81 .  x  =  Va  ■*-  (b  X  c) — Set  b  on  B"  to  a  on  A";  under  c  on  Br  read 

x  on  D. 

82.  x  =  axVbXc — Set  b  onBr  to  con  A;  at  a  on  C  readxon  D. 

83.  x  =  Va  X  b  -r  c — Set  c  on  C  under  a  on  A;  under  b  on  B"  read 

x  on  D. 

84.  x  =  a  Xb  X  Vc — Set  c  on  Br  over  a  on  D;  at  b  on  C  read  x 

on  D. 

85.  x  =  Va  Xbv  c — Set  c  on  C  to  b  on  D;  under  a  on  B"read  x 

on  D. 

86.  x  =  Va  -v-  ( Vb  X  c) — Set  b  on  BrI  under  a  on  A;  under  c  on 

CI  read  x  on  D. 

87.  x  =  Va3  Xb3  -T-  c3 — Set  a  on  Br  to  b  on  A';  under  c  on  B"  read 

x  on  E. 

88.  x  =  Va3-f-(b3Xc3")— Set  b  on  B"  to  a  on  A";  under  b  on  Br 

read  x  on  E. 

89.  x  =  Va3  Xb3  -s-  c3 — Set  c  on  C  under  a  on  A;  under  b  on  B" 

read  x  on  E. 

90.  x  =  a3  X  Vb3  +  c3 — Set  c  on  B"  to  b  on  A;  under  a  on  C  read 

x  on  E. 

91.  x=a3  X  b3  X  Vc3 — Set  a  on  CI  over  bon  D;  under  c  on 

BrI  read  x  on  E. 

92.  x  =  a3  X  Vb3  *  c3 — Set  c  on  C  to  a  on  D;  under  b  on  B"  read 

x  on  E. 

93.  x  =a3  Xb3-r-  Vc3 — Set  c  on  B"  over  a  on  D;  under  b  on  C 

read  x  on  E. 


PART  III. 
CONVERSION  RATIOS 


/ 


J 


Part  III — Conversion  Ratios 


I.     DECIMAL   EQUIVALENTS    OF  FRACTIONAL 
PARTS. 


FRAC- 
TION 

1 

328 

1 
64S 

DECIMAL 

FRAC- 
TION 

l 

1 
6iS 

DECIMAL 

1 

.015625 

33 

.515625 

1 

2 

.03125 

17 

34 

.53125 

3 

.046875 

35 

.546875 

A 

2 

4 

.0625 

Iff 

18 

36 

.5625 

5 

.078125 

37 

.578125 

3 

6 

.09375 

19 

38 

.59375 

7 

.109375 

39 

.609375 

t 

4 

8 

.125 

1 

20 

40 

.625 

9 

. 140625 

41 

.640625 

5 

10 

. 15625 

21 

42 

. 65625 

11 

.171875 

43 

.671875 

T3B 

6 

12 

.1875 

H 

22 

44 

.6875 

13 

.203125 

45 

.703125 

7 

14 

.21875 

23 

46 

.71875 

15 

.234375 

47 

.734375 

i 

8 

16 

.25 

f 

24 

48 

.75 

17 

.265625 

49 

.765625 

9 

18 

.28125 

25 

50 

.78125 

19 

.296875 

51 

.796875 

TB 

10 

20 

.3125 

H 

26 

52 

.8125 

21 

.328125 

53 

.828125 

11 

22 

. 34375 

27 

54 

.84375 

23 

. 359375 

55 

.859375 

I 

12 

24 

.375 

i 

28 

56 

.875 

25 

.390625 

57 

.890625 

13 

26 

.40625 

29 

58 

. 90625 

27 

.421875 

59 

.921875 

14 

28 

.4375 

H 

30 

60 

.  9375 

VB 

29 

.453125 

61 

.953125 

15 

30 

.46875 

31 

62 

.96875 

31 

.484375 

63 

.984375 

* 

16 

32 

.5 

l 

32 

64 

1. 

52 


Conversion  Ratios 


II.     METRIC  SYSTEM  OF  UNITS. 
MEASURES  OF  LENGTH,  CAPACITY  AND  WEIGHT. 

LENGTH 


KILO- 
METER 


HECTO- 
METER 


DECA- 
METER 


DECI- 
METER 


CENTI- 
METER 


MILLI- 
METER 


CAPACITY 

KILOLI- 

1    HECTO- 

DECALI- 

LITER 

TER  OR 

LITER   OR 

TER  OR 

OR 

DECILI- 

STERE 

DECI- 

CENTI- 

MILLI- 

TER 

1      STERE 

STERE 

8TERE 

CENTILI- 
TER 


MILLILITER 


WEIGHT 


KILO- 

HECTO- 

DECA- 

DECI- 

CENTI- 

GRAM 

GRAM 

GRAM 

GRAM 

GRAM 

GRAM 

MILLIGRAM 

1 

10 

100 

1,000 

10,000 

100,000  1,000,000 

.1 

1 

10 

100 

1,000 

10,000 

100,000 

.01 

.1 

1 

10 

100 

1,000 

10,000 

.001 

.01 

.1 

1 

10 

100 

1,000 

.000,1 

.001 

.01 

.1 

1 

10 

100 

.000,01 

.000,1 

.001 

.01 

.1 

1 

10 

.000,001 

.000,01 

.000,1 

.001 

.01 

.1 

1 

1  Myriameter  -  10  Kilometers  =  10,000  Meters. 
1  Metric  Ton  =  1,000  Kilograms  -  100  Quintals  =  10  Myria- 
grams. 

1  Liter  =  1  Cubic  Decimeter. 

SQUARE  OR  SURFACE  MEASURE. 


gf 

P5  w 

..  h  a 

H  H  < 

g 

«  g  « 

w  a 

■ 

"}  1 

28« 

«       <! 

«  g 

tf  H 

tf  w 

*§ 

g§« 

&d£ 

m  «  ° 

3  «  S 
P  H  5 

0D  H  « 

5  s 

<  s 

t>5 

CD  H 

gew 

CQ   y 

to  B 

©►J 

0Q  tJ 

M 

W  O 

ft 

S° 

fl 

i 

s 

1 

100 

10,000 
100 

1,000,000 

10,000 

100 

.01 

1 

i, 000,066 

10,000 

.000,1 

.01 

1 

1,000,000 

.000,001 

.000,1 

.01 

1 

100 

10,000 

1,000,666 

.000,001 

.000,1 

.01 

i 

100 

10,000 

.000,001 

.000,1 
.000,001 

.01 

1 

100 

.000,1 

.01 

1 

1  Square  Myriameter  =  100  Square  Kilometers  =  100,000,000 
Square  Meters. 

CUBIC  MEASURE. 


CUBIC 
DECAMETER 

CUBIC 
METER 

CUBIC 
DECIMETER 

CUBIC 
CENTIMETER 

CUBIC 
MILLIMETER 

1 

1000 

1 

.001 

.000,001 

000,000,001 

1,000,000 
1,000 

1 

.001 
.000,001 

1,000,000,000 

1,000,000 

1,000 

1 

.001 

.001 

.000,001 

.000,000,001 

1,000,000,000 

1,000,000 

1,000 

1 

1  Cubic  Meter  - 1  Kiloliter  =  1  Stere. 


Simple  Equivalents 


53 


III.     SIMPLE  EQUIVALENTS. 
GEOMETRICAL  RATIOS. 


SCALE    C 


SCALE    D 


Diameter  of  Circle 
Diameter  of  Circle 
Diameter  of  Circle 
Side  of  Square 
Area  of  Circle 
Area  of  Circle 


Circumference  of 

Circle 

Side  of  Inscribed 

Square 

Side     of     Equal 

Square 

Diagonal       of 

Square.  ...... 

Area  of  Inscribed 

Square 

Area  of  Circum- 
scribed Square. .  . 


SCALE 

SCALE 

IF    C—l, 

IF  D  =  l, 

C 

" 

D  = 

c  = 

226 

710 

3.1416 

.3183 

99 

70 

.7071 

1.4142 

79 

70 

.8862 

1 . 1284 

70 

99 

1.4142 

.7071 

300 

191 

.6366 

1 . 5708 

62 

79 

1.2732 

.7854 

LINE^ 

LR   MEASURE— (UNITED    STATES   AND 

BRITISH). 

INCHES 

FEET 

YARDS 

RODS 

FURLONGS 

MILES 

1 

12 

36 

198 

7,920 

63,360 

•083,33 

1. 

3- 

165 

660- 

5,280. 

.027,78 
.333,33 
1. 

5.5 
220. 
1,760. 

.005,050,5 
.060,606,1 
.181,818,2 
1. 
40. 
320. 

.000,126,26 
.001,515,15 
.004,545,45 
.025 

1. 

8. 

.000,015,78 
.000,189,39 
.000,568,18 
.003,125 
.125 
1. 

ROPE  AND  CABLE  MEASURE. 

1  Inch  =  .111,111  Span  =  .013,889  Fathom  =  .000,115,7 
Cable's  Length. 

1  Span  =  9  Inches  =  .125  Fathom  -  .001,041,67  Cable's 
Length. 

1  Fathom  =  6  Feet  =  8  Spans  =  72  Inches  =  .008,333 
Cable's  Length. 

1  Cable's  Length  =  120  Fathoms  =  720  Feet  =  960 
Spans    =  8,640  Inches. 

GUNTER'S  OR  SURVEYOR'S  CHAIN. 

1  Link=  7.92  Inches  =  .01  Chain  =  .000,125  Mile. 

1  Chain  =  100  Links  =  66  Feet  =  4  Rods  =  .012,5  Mile. 

1  Mile  =  80  Chains  =  8,000  Links. 

RATIOS  OF  LENGTHS. 


SCALE    C 

SCALE    D 

SCALE  c 

SCALE  D 

IF  C=l, 
D  = 

IPD  =  1, 
C  = 

1-64  Inch 

Millimeters. . . 
Millimeters. .  . 

Meters 

Meters 

Meters 

Meters 

Kilometers. .  . 
Knots 

320 

5 

82 

35 

37 

17 
23 

38 

127 
127 
25 
32 
186 

342 
37 
33 

.3969 

25.400 

.3048 

.9144 

5.0292 

20.117 
1 . 6094 
.8684 

2.5197 
. 03937 

Feet 

3.2808 

Yards 

1.0936 

Rods 

.1988 

Chains    (Survey- 
or's)  

Miles 

.0497 
.6214 

Miles 

1.1515 

54 


Conversion  Ratios 


SQUARE  OR  LAND  MEASURE— (UNITED  STATES 
AND  BRITISH). 


SQUARE    INCHES 

SQUARE    FEET 

SQUARE    YARDS 

1 

144 

1,296 

39,204 

6,272,640 

.006,944 

1. 

9. 
272.25 

43,560. 
27,878,400. 

.000,771,6 

.111,111 
1. 

30.25 
4,840. 
3,097,600. 

SQUARE    RODS 

ACRES 

SQUARE    MILES 

.033,06 

.000,206,6 
.006,25 
1. 
640. 

1. 

160. 
102,400. 

.000,009,77 
.001,562,5 
1. 

RATIOS  OF  AREAS. 


SCALE    C 

SCALE    D 

SCALE  c 

SCALE  D 

IF  C=l, 
D  = 

IF  D  =  l, 

c  = 

Circular  Mils .  .  . 

Square  Mils.  .  . 

79 

62 

.7854 

1.2732 

Circular  Inches 

Square    Centi- 

meters  

434 

2200 

5.0671 

.1974 

Square  Inches 

Square    Centi- 

meters  

31 

200 

6.4516 

.1550 

Square  Feet. .  .  . 

Square  Meters 

140 

13 

.0929 

10.764 

Square  Yards  .  . 

Square  Meters 

61 

51 

.8361 

1 . 1960 

Square  Miles 

Square      Kilo- 

meters  

56 

145 

2.590 

.3861 

Acres 

Hectares 

42 

17 

.4047 

2.471 

CUBIC    OR    SOLID    MEASURE— (UNITED    STATES    AND 
BRITISH). 

1  Cubic  Inch  =.000,578,7  Cubic  Foot  =.000,021,433  Cubic 
Yard. 

1  Cubic  Foot  =1,728  Cubic  Inches  =  .037,037,04  Cubic 
Yard. 

1  Cubic  Yard  =27  Cubic  Feet  =46,656  Cubic  Inches. 

1  Cord  of  Wood  =  128  Cubic  Feet  =4  Feet  by  4  Feet  by  8 
Feet. 

DRY  MEASURE— (UNITED  STATES  ONLY). 


PINTS 

QUARTS 

GALLONS    |       PECKS 

BUSHELS 

CUBIC   INCHES 

1 

2 

8 

16 

64 

.50 
1. 
4. 

8. 
32. 

.125 

.25 

1. 

2. 

8. 

.062,5 

.125 

.5 

1. 

4. 

.015,625 
.031,25 
.125 
.25 
1. 

33.600,312,5 
67.200,625 

268.802,5 

537.605 
2,150.42 

Simple  Equivalents 


55 


LIQUID  MEASURE— (UNITED  STATES  ONLY). 


GILLS 


PINTS 

QUARTS 

.25 

.125 

1. 

.5 

2. 

1. 

8. 

4. 

252. 

126. 

GALLONS 


BARRELS  CUBIC   INCHES 


1 

4 
8 
32 
2,008 


.031,25 
.125 
.25 
1. 
31.5 


.000,498 
.003,968 
.007,937 
.031,746 


1. 


7.218,75 
28.875 
57.75 
231. 
7,276.5 


RATIOS    OF    CAPACITIES. 


SCALE    C 

SCALE      D 

Cubic  Inches 

Cubic   Centime- 

ters   

Cubic  Feet.  . .  . 

Cubic  Meters.  .  . 

Cubic  Feet.  . .  . 

Bushels  (U.  S.) . 

Cubic  Yards. .  . 

Cubic  Meters.  .  . 

Pints  (U.  S. 

Pints  (U.  S. 

liquid) 

Dry) 

Pints  (U.  S. 

Liters 

Liquid) 

Pints  (U.  S. 

Dry)    

Gallons  (U.  S. 

Pints  (British) 

Liters 

Liquid)  .... 

Gallons  (U.  S. 

Imperial  Gal- 

Liquid)  

lons 

SCALE  c 

SCALE  D 

IF  C=l, 
D  = 

IPD  =  1, 
C  = 

36 

106 

61 

51 

590 

3 

49 

39 

16.387 
.0283 
,8036 
.7646 

.0610 
35.315 
1.2445 
1.3079 

71 

61 

.8594 

1 . 1637 

93 

44 

.4732 

2.1134 

96 

93 

.9690 

1.0320 

42 

159 

3.7854 

.2642 

6 

5 

.8327 

1.2009 

AVOIRDUPOIS  WEIGHT— (UNITED  STATES 
AND  BRITISH.) 


GRAINS 

DRAMS 

OUNCES 

1. 

27.343,75 
437.5 
7,000. 
784,000. 
5,680,000. 

.036,57 

1. 

16. 

256. 

28,672. 

573,440. 

.002,286 

.062,5 

1. 

16. 

1,792. 

35,840. 

POUNDS 

HUNDREDWEIGHTS 

LONG  OR  GROSS  TONS 

.000,143 
.003,906 
.062,5 
1. 
112. 
2,240. 

.000,001,28 

.000,034,88 
.000,558,04 
.008,928,6 
1. 
20. 

.000,000,176 
.000,001,744 
.000,027,90 
.000,446,4 
.05 
1. 

1  Pound  Avoirdupois  =  1.215,278  Pounds  Troy. 
1  Net  or  Short  Ton  =2,000  Pounds  =.892,857  Long  or  Gross 
Ton. 


TROY  WEIGHT— (UNITED   STATES   AND   BRITISH). 

GRAINS 

PENNYWEIGHTS 

OUNCES 

POUNDS 

1 
24 

480 
5,760 

.041,667 
1. 

20. 
240. 

.002,083,3 
.05 
1. 
12. 

.000,173,6 
.004,166,7 
.083,333,3 
1. 

56 


Conversion  Ratios 


APOTHECARIES'  WEIGHT— (UNITED  STATES  AND 
BRITISH). 


GRAINS 

SCRUPLES 

DRAMS 

OUNCES 

POUNDS 

1 

20 
60 

480 
5,760 

.05 
1. 
3. 
24. 

288. 

.016,667 
.333,333 

1. 

8. 
96. 

.002,083,3 
.041,666,7 
.125 
1. 
12. 

.000,173,611 
.003,472,2 
.010,416,7 
.083,333,3 
1. 

The  pound,  ounce  and  grain  are  the  same  as  in  troy  weight. 
The  avoirdupois  grain  =  troy  grain  =  apothecaries'  grain. 


SCALE    C 

SCALE      D 

SCALE 

c 

SCALE 
D 

IF  C  = 
1,D  = 

IP  D  = 

1,  c  = 

Grains 

Grams 

710 
6 
90 
97 
93 
75 
62 

46 

170 
82 
44 
83 
68 
63 

i    .0648 
28.350 
.9115 
.4536 
.8929 
.9072 
1.0161 

15.432 

Ounces  (Avoird.).  .  . 

Grams 

.  0353 

Ounces  (Avoird.).  .  . 
Pounds  (Avoird.) . .  . 

Tons  (Short) 

Tons  (Short) 

Tons  (Long) 

Ounces  (Troy) 

Kilograms 

Tons  (Long) 

Metric  Tons 

Metric  Tons 

1.0971 
2.2046 
1.1200 
1 . 1023 
.9842 

ANGULAR  OR  CIRCULAR  MEASURE. 


SECONDS 

MINUTES 

DEGREES 

RADIANS 

QUADRANTS 

CIRCUM- 
FERENCES 

1 

.016,67 

1 

60 

3,437.8 

5,400 

21,600 

.000,278 

.016,67 

1. 

57.296 

90. 

360. 

.000,004,9 

.000,291 

.017,453 

1. 

1.570,8 

6.283,2 

.000,003,1 
.000,185 
.011,111 
.636,62 

1. 

4. 

60 

3,600 

206,265 

324,000 

.000,046,3 
.002,777,8 
.159,155 
.25 
1. 

TIME— (MEAN  SOLAR). 


SEC- 
ONDS 

MIN- 
UTES 

HOURS 

DAYS 

WEEKS 

MONTHS 

(aver- 
age) 

YEARS 

(365 

days) 

1 

60 

.016,67 

1 

60 

1,440 

10,080 

43,800 

525,600 

.000,278 

.016,67 

1 

r24 

168 

730 

8,760 

.000,011,6 
.000,694,4 
.041,667 
1. 
7. 
30.417 
365. 

'.  666,099,2 

.005,952,4 
.  142,86 

1. 

4.345,24 
52 . 143 

3,600 

86,400 

604,800 

.001,37 
.032,88 
.230,14 
1. 
12. 

.000,114 
.002,74 
.019,18 
.083,33 

1. 

PAPER  MEASURE. 

SHEETS 

QUIRES 

REAMS 

1 

24 

480 

.041,67 
1. 
20 

.002,08 
.05 
1. 

Compound  Equivalents 


57 


TABLE  OF  UNITS. 


UNITS                          DOZEN 

GROSS 

GREAT   GROSS 

1 
12 

144 

1,728 

.083,33 
1. 
12. 
144. 

.006,944 
.083,33 
1. 
12. 

.000,578,7 
.006,944 
.083,33 
1. 

MONEY  EQUIVALENTS— (GOLD  BASIS). 

DOLLARS 

POUNDS 

FRANCS 

MARKS 

RUBLES 

PESOS 

100 

20 

100 

100 

100 

100 

PESOS 

CENTS 

SHIL- 
LINGS 

CEN- 
TIMES 

PFENNIG 

KOPECKS 

CENTAVOS 
MEXICO 

U.  S.  AND 

GREAT 

GER- 

RUSSIA 

(silver 

CUBA 

CANADA 

BRITAIN 

FRANCE 

MANY 

stan'd) 

1. 

.2055 

5.18 

4.20 

1.942 

2.611 

1.080 

4.8665 

1. 

25.21 

20.44 

9.451 

12.706 

5.256 

.193 

.0396 

1. 

.811 

.375 

.504 

.208 

.238 

.0489 

1.233 

1. 

.462 

.621 

.257 

.515 

.1058 

2.668 

2.163 

1. 

1.345 

.556 

.383 

.0787 

1.984 

1.609 

.744 

1. 

.414 

.926 

.1901 

4.797 

3.889 

I     1.798 

2.418 

1. 

IV.     COMPOUND  EQUIVALENTS. 
VELOCITIES. 


Feet  per  Second. 
Feet  per  Second. 
Feet  per  Second. 
Feet  per  Second. 
Feet  per  Minute. . 
Miles  per  Hour.  . 
Knots  per  Hour  . 


SCALE    D 

Meters  per  Second.  . 

Miles  per  Hour 

Knots  per  Hour.  .  .  . 
Kilometers  per  Hour 

Miles  per  Hour 

Meters  per  Second.  . 
Kilometers  per  Hour 


SCALE 

SCALE 

IF  C  = 

C 

D 

1,  D  = 

82 

25 

.3048 

22 

15 

.6818 

76 

45 

.5921 

41 

45 

1.0973 

264 

3 

.0114 

85 

38 

.4470 

34 

63 

1.8533 

IF  D  = 

l,c  = 


3.2808 
1.4667 
1 . 6889 

.9113 
88. 
2.2369 

.5396 


WEIGHTS,  LENGTHS,  ETC. 


SCALE    C 

SCALE    D 

SCALE 
C 

SCALE 
D 

IF  c  = 

1,D  = 

IF  D  = 
1,  C  = 

Pounds  (Avoird.) 
per  Mile 

Pounds  (Avoird.) 
per  Mile 

Pounds  (Avoird.) 
per  Yard 

Pounds  per  Foot 
(Steel) 

Pounds  per  Yard 
(Steel) 

Pounds  per  Yard  . 

Circular  Mils  (Cop- 

Grains  per  inch 
Kilograms   per   Kilo- 

190 
540 

230 
630 

500 

14 

18,500 
125 

21 
152 

114 

185 

49 
11 

56 

2 

.1105 
.2818 

.4961 
.2939 

.0980 

.7857 

. 00303 

9.0514 
3 . 5480 

Kilograms  per  Meter 

Square  Inches  Section 

Square  Inches  Section 

Tons  (Long)  per  Mile . 
Pounds  per  1000  Feet 

Pounds  per  Mile 

2.0159 

3.4017 

10.205 
1.2727 

330.3 

Circular  Mils  (Cop- 
per)  

.01598 

62.56 

58 


Conversion  Ratios 
CAPACITIES  AND  WEIGHTS. 


Pounds  per  Cu. 

Inch 

Pounds  per  Cu. 

Foot 

Pounds  per  Cu. 

Foot 

Pounds  per  Cu. 

Foot 

Pounds  per  Cu. 

Foot 

Tons  (Short)  per 

Cu.  Yard 


Grams  per  Cu.  Centi- 
meter  

Grains  per  Cu.  Inch 

Pounds  per  Gallon 

(U.  S.  Liquid) 

Tons   (Short)  per  Cu. 

Yard ! 

Kilograms     per      Cu. 

Meter J 

Metric  Tons    per  Cu.1 

Meter I 


SCALE 

c 

SCALE 
D 

6 

166 

40 

162 

30 

4 

2,000 

27 

181 

2,900 

75 

89 

IF  C  = 
1,  D  = 


27.680 
4.0509 


IF  D  = 

1.  c- 


.0361 
.2469 
.1337  7.4805 
.0135  74.074 


16.018 
1 . 1866 


.0624 
.8428 


PRESSURES. 


Pounds  per  Sq. 

Inch 

Pounds  per  Sq. 

Inch 

Pounds  per  Sq. 

Inch 

Pounds  per  Sq. 

Inch 

Pounds  per  Sq. 

Inch 

Pounds  per Sq. 

Inch 

Pounds  per  Sq. 

Inch 

Pounds  per  Sq. 

Foot 

Tons  (Short)  per 

Sq.  Foot 

Feet  of  Water 

Feet  of  Water.  .  . 
Inches  of  Mercury. 


Tons  (Short)  per  Sq 
Foot 

Tons  (Long)  per  Sq 
Foot 

Metric  Tons  per  Sq. 
Meter 

Atmospheres 

Feet  of  Water 

Millimeters  of  Mercu- 
ry  

Inches  of  Mercury 


Sq. 


Kilograms    per 

Meter 

Metric  Tons    per  Sq. 

Meter 

Pounds  per  Sq.  Foot; . 

Atmospheres 

Atmospheres 


1,000 

700 

64 

485 

52 

89 

56 

43 

43 
25 

610 
4,100 


SCALE 

'  IF  C  = 

IF  D  = 

D 

1,D  = 

1,C  = 

72 

.0720 

13.889 

45 

.0643 

15.556 

45 

.7031 

1.4223 

33 

.0680 

14.697 

120 

2.3067 

.4335 

4,600 

51.712 

.01934 

114 

2.0359 

.4912 

210 

4.8824 

2.048 

420 

1,560 

18 

137 

9.7648 

62.428 

.0295 

1    .0334 

.1024 

.0160 

33.901 

29.921 

WORK. 


Fool  -pounds .  . 
Foot-pounds.  . 
Foot-pounds.  . 
Horse  -  power 

Hours  .  . 
Horse  -  power 

Hours 

B.  T.  U 

B.  T.  U 

B.  T.  U 

B.  T.  U 

B.  T.  U. 

B.  T.  U 


SCALE    D 

SCALE 
C 

SCALE 
D 

IF  C  = 
1,  D  = 

IF  D  = 
1,  C  = 

45 

68 

600 

59 

92 

9 

18 

690 

41 

23 

58 

61 

22 

83 

44 

59 

7,000 

19 

174 

58 

33 

17 

1 . 3557 
.3239 
.1383 

.7457 

.6412 
778.1 

1.0549 
.2520 

1.4147 

1.4344 
.2930 

.7376 

Calories  (Small) 

Kilogram  Meters 

Kilowatt  Hours 

Calories  (Small) 

Foot-pounds 

Kilowatt  Seconds .... 

Calories  (Large) 

Horse-power  Seconds 
Metric     Horse-power 

3.0878 
7.2330 

1.3411 

1.5595 
.001285 

.9480 
3 . 9683 

.7068 

.6972 

Watt  Hours 

3.4127 

Compound  Equivalents 


59 


MONEY,  LENGTH,  WEIGHT,  ETC. 


SCALE    C 

Cents  per  Ounce 

( Avoir  d.) 

Dollars  per  Ton 

(Short) 

Dollars  per  Ton 

(Long) 

Dollars  per  Pound 

(Avoird.) 

Cents  per  Yard.  .  . 
Dollars  per  Mile. .  . 
Dollars  per  Mile. .  . 


Dollars    per     Pound 

(Avoird.) 

Francs  per  MetricTon 

Francs  per  Metric  Ton 

Marks  per  Kilogram 

Francs  per  Meter.  .  .  . 

Shillings  per  Mile 

Francs  per  Kilometer 


SCALE     SCALE       IF  C  =  I   IF  D  = 
1,D=      1,  C- 


100 

10 

10 

108 

1000 

73 

59 


16 

57 

51 

1,000 

56 

300 

190 


.160 

5.70 

5.10 

9.26 
.056 
4.11 
3.22 


6.25 

.175 

.196 

.108 

17.64 

.243 

.311 


Price  List  of  E.  D.  Co.'s  Slide  Rules 


No.  1762A.  Multiplex  Slide  Rule,  with  Cube  and  Recip- 
rocal Scales,  5  in.,  Boxwood  or  Mahogany, 
divisions  on  white  ivorine,  with  glass  indicator 
and  our  Patented  Adjustment;  printed  in- 
structions with  rule Each,     $5  00 

1762B.  Multiplex  Slide  Rule,  with  Cube  and  Recip- 
rocal Scales,  10  in.,  Boxwood  or  Mahogany, 
divisions  on  white  ivorine,  with  glass  indicator 
and  our  Patented  Adjustment;  printed  in- 
structions with  Rule Each,       5  00 

1762C.  Multiplex  Slide  Rule,  with  Cube  and  Recipro- 
cal Scales,  20  in.,  Boxwood  or  Mahogany, 
divisions  on  white  ivorine,  with  glass  indicator 
and  our  Patented  Adjustment;  printed  in- 
structions with  rule Each,     15100 

1763A.  Multiplex  Slide  Rule,  with  Reciprocal  Scale 
but  no  Cube  Scale,  5  in.,  Boxwood  or  Mahog- 
any ;  divisions  on  white  ivorine,  with  glass  in- 
dicator and  our  Patented  Adjustment;  print- 
ed instructions  with  Rule Each,       4  50 

1763B.  Multiplex  Slide  Rule,  with  Reciprocal  Scale  4 
but  no  Cube  Scale,  10  in.,  Boxwood  or  Mahog- 
any; divisions  on  white  ivorine,  with  glass 
indicator    and    our    Patented    Adjustment; 
printed  instructions  with  Rule Each,       4  50 

1763C.  Multiplex  Slide  Rule,  with  Reciprocal  Scale 
but  no  Cube  Scale,  20  in.,  Boxwood  or  Mahog- 
any; divisions  on  white  ivorine,  with  glass 
indicator  and  our  Patented  Adjustment; 
printed  instructions  with  Rule Each,     14  50 

1764.  The  Mack  Improved  Slide  Rule  (Mannheim), 
5  in.,  Hardwood,  divisions  on  white  ivorine, 
with  glass  indicator ;  printed  instructions  with 

Rule Each,       4  50 

1765.  The  Mack  Improved  Slide  Rule  (Mannheim), 
10  in.,  Hardwood,  divisions  on  white  ivorine, 
with  glass  indicator ;  printed  instructions  with .... 

Rule Each,       4  50 

1767.  The  Mack  Improved  Slide  Rule  (Mannheim), 
20  in.,  Hardwood,  divisions  on  white  ivorine, 
with  glass  indicator ;  printed  instructions  with 

Rule Each,     14  50 

1768.  Book  with  complete  instructions  for  the  use 
of  Slide  Rules  free  of  charge  if  ordered  with 
1762-1777,  otherwise Each,  50 

1769.  Glass  Indicator  for  Slide  Rule Each,  75 

1770.  The  Standard  Adjustable  Slide  Rule  (Mann- 
heim), 5  in.,  divisions  on  white  ivorine,  with 

glass  indicator  and  printed  instructions Each,       4  50 

1771.  The  Standard  Adjustable  Slide  Rule  (Mann- 
heim), 10  in.,  divisions  on  white  ivorine,  with 

glass  indicator  and  printed  instructions Each,       4  50 

1772.  The  Standard  Adjustable  Slide  Rule  (Mann- 
heim), 20  in.,  divisions  on  white  ivorine,  with 

glass  indicator  and  printed  instructions Each,     14  50 

1776.  The  Union  Slide#  Rule  (Mannheim),  5  in., 
divisions  on  white  ivorine,  with  glass  indicator; 

printed  instructions  with  Rule Each,       3  30 

1777.  The  Union  Slide  Rule  (Mannheim),  10  in., 
divisions  on  white  ivorine,  with  glass  indicator; 

printed  instructions  with  Rule Each,       3  60 

1787.     The  Engineers'  Slide  Rule,  24  in.,  Hardwood, 

with  directions Each,       5  00 

1794.  Fuller's  Spiral  Slide  Rule,  in  Mahogany  Box, 

with  directions Each,     30  00 

1795.  College  Slide  Rule  (Mannheim),  10  in.,  Hard- 
wood, divisions  on  white  paper,  with  glass  in- 
dicator and  directions Each,       1  25 


Eugene   Dietzgen  Co. 


181    Monroe   Street 
Chicago,  III. 

14    First   Street 
San  Francisco,  Cal. 


1 19-12 1  W.  23d  Street 
New  York,  N.  Y. 

145   Baronne  Street 
New    Orleans,    La. 


manufacturers  and  importers  of 

Surveying  and  Engineering 
Instruments 

MATHEMATICAL  INSTRUMENTS 
DRAFTING  SCALES,  PROTRACT- 
ORS,   TRIANGLES,     T     SQUARES 


CURVES,  DRAWING  BOARDS  AND  TABLES, 
INDIA  INKS,  THUMB  TACKS,  BRUSHES, 
CHINA  WARE,  KOH  -  I  -  NOOR  PENCILS, 
RUBBERS,  TAPES,  CHAINS,  LEVELING 
RODS,    ANEROIDS,    ETC. 


Complete    Illustrated    Catalogue    and    Price    List 
Mailed   on  Application   to   any  Dealer  or     fc 
Professional   of    Good   Standing. 


14  DAY  ITSP 

LOAN  OEPT 

_____wto  immediate  recall. 


LD  2lA-40m-ll  '63 

(El602slO)476B 


Pamphlet 
Binder 
Gaylord  Bros.,  Inc.] 

Stockton,  Calif. 
T.M.  Reg.  U.S.  Pat.  Off.  I 


M92099 


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